#6494125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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	#37553	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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:

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

Polycyclic presentation:

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

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, 6401321);