#7336721

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7243917);