#382462

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2470 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 2470 conjugacy classes of subgroups — too many to draw. Only its 362 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,d^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,d^c=de

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289658);