#6498449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6405645);