#124924

order	512
nilpotency class	4
isoclinism	isoclinic class of #124865
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1525
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#376	C₂³
socle	C₂	#7639
prime radical	C₂³	#410
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,13,16,37,17,38,27,55)(4,18,21,48)(5,23,25,56)(6,11)(7,12)(8,15,29,44,31,24,30,45)(9,22,33,49)(10,26,34,50)(14,39,41,61,43,60,42,36)(19,35)(20,51)(28,57,58,40,59,62,46,52)(32,54)(47,63)(53,64);(1,3,7,17)(2,8)(4,14,21,43)(5,24,25,15)(6,27,19,16)(9,28)(10,13)(11,29)(12,31)(18,46)(20,52,54,40)(22,41,49,42)(23,55)(26,45,50,44)(30,35)(32,36)(33,59)(34,38)(37,56)(39,47)(48,58)(51,61)(53,62,63,57)(60,64);(1,4,19,49,7,21,6,22)(2,9,11,18,12,33,35,48)(3,14,16,42,17,43,27,41)(5,20,50,63,25,54,26,53)(8,28,29,46,31,59,30,58)(10,32,23,47,34,51,56,64)(13,36,55,39,38,61,37,60)(15,40,45,62,24,52,44,57)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgh,d^2=gi,e^2=h,f^2=i,b^a=bd,c^a=cfh,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32120);