#6491653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 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:

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

Polycyclic presentation:

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

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, 6398849);