#125324

order	512
nilpotency class	4
isoclinism	isoclinic class of #125098
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#466	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

461 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 94 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)(17,18,21,25,23,26,31,32,24,27,20,30,22,19,28,29);(1,3)(2,5)(4,8)(6,7)(10,13)(11,16)(14,15)(17,19)(18,22)(20,25)(21,30)(23,27)(24,26)(28,29)(31,32);(2,6)(5,7)(10,14)(13,15)(17,20,23,28,24,21,22,31)(18,25,26,32,27,30,19,29)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfi,d^a=df,c^b=ch,d^b=dg,e^b=efg,g^b=gi,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32520);