#6477649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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	#37455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,28)(24,34)(25,30)(26,31)(27,32)(29,35)(33,36);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,28)(24,27)(25,35)(26,33)(29,30)(31,36)(32,34);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,23)(22,29,32,36)(25,35,26,33)(28,31,34,30);(7,14)(10,18)(13,15)(17,19)(21,25,27,26)(22,30,32,31)(23,33,24,35)(28,36,34,29);(6,12)(10,18)(11,20)(17,19)(21,25,27,26)(22,31,32,30)(23,33,24,35)(28,29,34,36)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi

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, 6384845);