#6685237

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	#29738	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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)(13,14,17,21)(15,23,25,19)(16,27,28,43)(18,31,30,22)(20,34,35,29)(24,38,39,42)(26,33,41,37)(32,40,36,44);(1,2)(3,4)(6,9)(7,11)(13,15,17,25)(14,19,21,23)(16,24,28,39)(18,26,30,41)(20,32,35,36)(22,33,31,37)(27,42,43,38)(29,40,34,44);(2,4)(13,16,18,29,17,28,30,34)(14,20,22,27,21,35,31,43)(15,24,26,40,25,39,41,44)(19,32,33,42,23,36,37,38);(6,9)(10,12)(14,21)(15,25)(20,35)(22,31)(24,39)(26,41)(27,43)(40,44);(2,4)(5,7,8,11)(6,10,9,12)(15,25)(19,23)(24,39)(26,41)(32,36)(33,37)(38,42)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh

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, 6592433);