#6675043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6582239);