#6570616

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30274	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2090 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:

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

Polycyclic presentation:

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

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, 6477812);