#6500075

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3231 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:

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

Polycyclic presentation:

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

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, 6407271);