#331076

order	512
nilpotency class	3
isoclinism	isoclinic class of #331055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10280	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ei,d^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=de,e^c=eh,f^c=fh,g^c=gi,g^d=gh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238272);