#380740

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1134 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 1134 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287936);