#6584249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	C₂×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2093 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491445);