#6545212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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, 6452408);