#238461

order	512
nilpotency class	3
isoclinism	isoclinic class of #232313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1012 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145657);