#6440006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,20,26,35)(16,28,30,36)(17,22,18,31)(21,32,37,29)(24,38,40,43)(25,34,27,41)(33,42,44,39);(1,2)(3,4)(6,9)(10,12)(13,15,19,26)(14,20,23,35)(16,24,30,40)(17,25,18,27)(21,33,37,44)(22,34,31,41)(28,38,36,43)(29,39,32,42);(2,4)(13,16,19,30)(14,21,23,37)(15,24,26,40)(17,29,18,32)(20,33,35,44)(22,36,31,28)(25,39,27,42)(34,43,41,38);(2,4)(13,17,19,18)(14,22,23,31)(15,25,26,27)(16,29,30,32)(20,34,35,41)(21,36,37,28)(24,39,40,42)(33,43,44,38);(5,7,8,11)(6,10,9,12)(15,26)(16,30)(20,35)(21,37)(25,27)(28,36)(29,32)(34,41)

Polycyclic presentation:

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

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, 6347202);