#122876

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×#331
prime radical	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

251 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=fgi,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30072);