#405009

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

908 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fi,c^2=e,d^2=gi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312205);