#122938

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×#335
prime radical	C₂³	#335
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

154 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,12)(4,9)(5,19)(6,21)(7,11)(8,15)(10,13)(14,25)(16,31)(17,27)(18,24)(20,32)(22,29)(23,30)(26,28)(33,34,36,40,37,41,46,52,38,42,47,53,48,54,58,62)(35,39,44,50,45,51,57,61,43,49,55,59,56,60,63,64);(1,3,13,28,6,16,20,30,7,17,18,29,4,14,5,15)(2,8,19,25,9,22,24,27,11,23,32,31,21,26,10,12)(33,35)(34,39)(36,44)(37,45)(38,43)(40,50)(41,51)(42,49)(46,57)(47,55)(48,56)(52,61)(53,59)(54,60)(58,63)(62,64);(1,4,7,6)(2,9,11,21)(3,14,17,16)(5,18,20,13)(8,22,23,26)(10,19,24,32)(12,25,27,31)(15,29,30,28)(35,43)(39,49)(44,55)(45,56)(50,59)(51,60)(57,63)(61,64)

Polycyclic presentation:

2^9|a^2=e,b^2=df,c^2=h,d^2=fh,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ci

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30134);