#124469

order	512
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	C₂³	C₂×#104
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 320 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210337, #72210748
maximal normal subgroup of	#72210337
maximal quotient of	#72210317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31665);