#6445020

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#24784	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2050 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)(13,14,17,19)(15,23,26,36)(16,21,28,32)(18,30,24,22)(20,33,31,27)(25,37,39,44)(29,41,38,35)(34,43,42,40);(1,2)(3,4)(6,9)(10,12)(14,19)(16,28)(20,31)(22,30)(23,36)(25,39)(29,38)(40,43);(2,4)(13,15,24,33,17,26,18,27)(14,20,30,36,19,31,22,23)(16,25,38,43,28,39,29,40)(21,34,41,44,32,42,35,37);(2,4)(13,16)(14,21)(15,25)(17,28)(18,29)(19,32)(20,34)(22,35)(23,37)(24,38)(26,39)(27,40)(30,41)(31,42)(33,43)(36,44);(5,7,8,11)(6,10,9,12)(16,28)(21,32)(25,39)(29,38)(34,42)(35,41)(37,44)(40,43)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,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, 6352216);