#7181124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7088320);