#6440805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2695 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)(13,14,19,25)(15,26,31,40)(16,32,34,41)(17,22,37,38)(18,23,27,24)(20,29,39,30)(21,28,42,33)(35,43,36,44);(1,2)(3,4)(6,9)(10,12)(13,15,27,29,19,31,18,30)(14,20,23,40,25,39,24,26)(16,17,28,35,34,37,33,36)(21,38,41,43,42,22,32,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,33,19,34,27,28)(14,21,24,32,25,42,23,41)(15,17,30,36,31,37,29,35)(20,38,26,44,39,22,40,43);(2,4)(13,17)(14,22)(15,28)(16,30)(18,36)(19,37)(20,32)(21,40)(23,44)(24,43)(25,38)(26,42)(27,35)(29,34)(31,33)(39,41);(13,18,19,27)(14,23,25,24)(15,29,31,30)(16,33,34,28)(17,35,37,36)(20,26,39,40)(21,41,42,32)(22,43,38,44)

Polycyclic presentation:

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

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, 6348001);