#129176

order	512
nilpotency class	4
isoclinism	isoclinic class of #129125
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1776
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8287
prime radical	C₂³	#444
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

211 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,13,16,43,18,44,17,38)(4,19,23,54)(5,25)(6,12)(7,11)(8,15,31,49,32,50,33,26)(9,24,37,61)(10,28)(14,45,47,63,20,55,51,42)(21,41)(22,58,27,53)(29,40)(30,59,52,46,34,62,64,48)(35,57)(36,56,39,60);(1,3)(2,8,11,32)(4,20,23,14)(5,26)(6,16)(7,18)(9,34)(10,38,25,43)(12,31,40,33)(13,41,44,35)(15,21)(17,29)(19,52)(22,59,27,62)(24,47,61,51)(28,49)(30,37)(36,45)(39,55)(42,53)(46,60,48,56)(50,57)(54,64)(58,63);(1,4,21,56,29,61,5,27,7,23,57,60,6,24,28,22)(2,9,35,58,40,54,10,39,11,37,41,53,12,19,25,36)(3,14,15,48,17,47,26,59,18,20,50,46,16,51,49,62)(8,30,13,42,33,52,38,45,32,34,44,63,31,64,43,55)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi,c^b=ch,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224946, #72224947, #72224954, …
maximal normal subgroup of	#72224946, #72224947, #72224954, …
maximal quotient of	C₂×#129176

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36372);