#7104733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7011929);