#303857

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

891 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,7)(4,6)(5,8)(10,14)(11,13)(17,18)(19,23)(20,30)(21,26)(22,27)(24,29)(25,31)(28,32);(1,3,5,4)(2,6,8,7)(9,10,12,14)(11,16,13,15)(17,19,22,29)(18,23,27,24)(20,31,28,21)(25,32,26,30);(1,3,5,4)(2,7,8,6)(10,15,14,16)(17,20,31,19,22,28,21,29)(18,24,26,32,27,23,25,30);(1,4,5,3)(2,7,8,6)(9,11,12,13)(10,16,14,15)

Polycyclic presentation:

2^9|b^2=eh,c^2=efhi,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, 211053);