#7167124

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	#30995	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,30)(23,31,35,47)(24,28,37,44)(25,39,32,29)(27,40,42,34)(33,46,36,51)(38,52,48,45)(41,49,43,50);(7,13)(8,16)(10,17)(11,15)(21,23,32,40,26,35,25,34)(22,27,39,47,30,42,29,31)(24,36,48,50,37,33,38,49)(28,43,52,46,44,41,45,51);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24,26,37)(22,28,30,44)(23,33,35,36)(25,38,32,48)(27,41,42,43)(29,45,39,52)(31,46,47,51)(34,50,40,49);(7,13)(10,17)(14,19)(18,20)(23,34)(24,37)(25,32)(27,31)(28,44)(29,39)(33,49)(35,40)(36,50)(41,51)(42,47)(43,46);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,35)(24,37)(27,42)(28,44)(31,47)(34,40)(38,48)(45,52)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=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, 7074320);