#297034

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 791 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=fi,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204230);