#7182580

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

1698 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)(21,22,27,29,26,32,25,31)(23,28,38,43,33,42,35,40)(24,30,39,44,36,46,37,45)(34,41,49,51,47,52,48,50);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,28)(24,34)(25,35)(26,33)(27,38)(29,43)(30,41)(31,40)(32,42)(36,47)(37,48)(39,49)(44,51)(45,50)(46,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(22,29)(23,33)(24,36)(25,27)(28,40)(30,45)(31,32)(41,51)(42,43)(44,46)(48,49)(50,52);(2,4)(7,14)(10,13)(15,20)(18,19)(21,24,26,36)(22,30,32,46)(23,34,33,47)(25,37,27,39)(28,41,42,52)(29,44,31,45)(35,48,38,49)(40,50,43,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,33)(24,36)(28,42)(30,46)(35,38)(37,39)(40,43)(44,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,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, 7089776);