#6506753

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2989 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=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, 6413949);