#297022

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 746 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)(10,14)(11,13)(17,18,22,26)(19,23,28,31)(20,29,27,32)(21,25,30,24);(2,5)(6,7)(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,30)(21,27)(22,28)(24,29)(25,32)(26,31);(10,15,14,16)(17,20,21,19,22,27,30,28)(18,23,24,29,26,31,25,32);(1,3,4,8)(2,6,5,7)(9,11,12,13)(10,16,14,15)

Polycyclic presentation:

2^9|a^2=ehi,c^2=f,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, 204218);