#7182568

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7089764);