#317998

order	512
nilpotency class	3
isoclinism	isoclinic class of #317982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

523 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225194);