#6607131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2249 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)(5,6)(7,10)(8,11)(9,12)(13,14,19,24)(15,25,27,36)(16,30,31,38)(17,23,18,22)(20,29,35,28)(21,32,37,33)(26,34,40,44)(39,41,43,42);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,28,19,27,18,29)(14,20,22,25,24,35,23,36)(16,26,32,41,31,40,33,42)(21,34,38,39,37,44,30,43);(7,12)(9,10)(13,16,18,33,19,31,17,32)(14,21,23,30,24,37,22,38)(15,26,29,42,27,40,28,41)(20,34,36,43,35,44,25,39);(5,7)(6,10)(8,12)(9,11)(15,27)(20,35)(25,36)(26,40)(28,29)(34,44)(39,43)(41,42);(7,12)(9,10)(15,27)(16,31)(20,35)(21,37)(25,36)(28,29)(30,38)(32,33)

Polycyclic presentation:

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

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, 6514327);