#6500473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2656 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407669);