#6859959

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,36)(28,40)(29,34)(30,32)(31,38)(33,42)(35,44)(37,46)(39,47)(41,49)(43,50)(45,52)(48,54)(51,55)(53,56);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,38)(26,31,34,36)(28,41,42,37)(30,39,44,52)(32,45,35,47)(33,49,40,46)(43,53,54,51)(48,56,50,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,44)(31,45)(33,48)(34,35)(36,47)(39,52)(40,50)(43,54)(46,55)(49,56)(51,53);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28,29,42)(26,33,34,40)(27,37,38,41)(30,43,44,54)(31,46,36,49)(32,48,35,50)(39,51,52,53)(45,55,47,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(27,38)(28,42)(31,36)(33,40)(39,52)(43,54)(45,47)(48,50)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6767155);