#6673555

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667700
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580751);