#124378

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂×#121	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 403 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,7,8,5)(4,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,24,22,25,29,31,21,23,19,26,27,30,28,32);(1,3)(2,5)(4,8)(6,7)(10,13)(11,16)(14,15)(18,23)(19,20)(25,30)(28,29);(2,6)(5,7)(10,14)(13,15)(17,19,22,28,21,20,27,29)(18,24,25,31,23,26,30,32)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cf,d^a=df,c^b=ci,d^b=dg,e^b=efg

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, 31574);