#7236518

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7143714);