#6558062

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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×#3271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2497 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)(13,14,18,22)(15,23,26,39)(16,20,29,36)(17,31,24,21)(19,32,34,27)(25,38,28,42)(30,44,40,37)(33,43,35,41);(5,6)(7,10)(8,9)(11,12)(13,15,24,32,18,26,17,27)(14,19,31,39,22,34,21,23)(16,28,40,41,29,25,30,43)(20,35,44,38,36,33,37,42);(1,2,3,4)(6,9)(10,12)(13,16)(14,20)(15,25)(17,30)(18,29)(19,33)(21,37)(22,36)(23,38)(24,40)(26,28)(27,41)(31,44)(32,43)(34,35)(39,42);(5,7)(6,10)(8,11)(9,12)(15,26)(16,29)(19,34)(20,36)(23,39)(27,32)(30,40)(37,44);(7,11)(10,12)(16,29)(20,36)(25,28)(30,40)(33,35)(37,44)(38,42)(41,43)

Polycyclic presentation:

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

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, 6465258);