#7182495

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(21,22,25,29,26,30,38,44)(23,31,33,40,34,43,37,27)(24,28,36,42,35,41,48,51)(32,45,47,50,46,52,49,39);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,26,34)(22,27,30,40)(24,32,35,46)(25,37,38,33)(28,39,41,50)(29,43,44,31)(36,49,48,47)(42,52,51,45);(2,4)(7,14)(10,13)(15,20)(18,19)(21,24,26,35)(22,28,30,41)(23,32,34,46)(25,36,38,48)(27,39,40,50)(29,42,44,51)(31,45,43,52)(33,47,37,49);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(24,35)(28,41)(32,46)(36,48)(39,50)(42,51)(45,52)(47,49)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7089691);