#382466

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2994 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 2994 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) = 64

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289662);