#128806

order	512
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1514
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8216
prime radical	#3054	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280909, #72281057, #72281216
maximal normal subgroup of	#72280909, #72281057, #72281216

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36002);