#7182527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1641 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)(21,22,25,29,26,30,39,45)(23,31,33,41,34,44,38,27)(24,35,37,51,36,28,42,43)(32,47,49,40,48,46,52,50);(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)(22,27,30,41)(24,32,36,48)(25,38,39,33)(28,40,35,50)(29,44,45,31)(37,52,42,49)(43,47,51,46);(2,4)(7,14)(10,13)(15,20)(18,19)(21,24,26,36)(22,28,30,35)(23,32,34,48)(25,37,39,42)(27,40,41,50)(29,43,45,51)(31,46,44,47)(33,49,38,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(24,36)(28,35)(32,48)(37,42)(40,50)(43,51)(46,47)(49,52)

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,d^a=di,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089723);