#375733

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1074 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 306 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=efh,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282929);