#7125706

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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^c=ei,e^d=ei,f^d=fi

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, 7032902);