#304996

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

708 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8)(3,7,4,6)(9,10,14,11)(12,19,21,20)(13,17,24,16)(15,23,18,22)(25,26,30,32)(27,31,35,40)(28,29,37,39)(33,34,36,38);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,20,21,19)(13,17,24,16)(15,23,18,22)(25,27,30,35)(26,31,32,40)(28,36,37,33)(29,38,39,34);(1,3,5,4)(2,7,8,6)(9,12,13,22,14,21,24,23)(10,15,16,19,11,18,17,20)(25,28,29,32,30,37,39,26)(27,33,34,40,35,36,38,31);(1,4,5,3)(2,7,8,6)(12,21)(15,18)(19,20)(22,23)

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=ef,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,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212192);