#7307624

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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

1461 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)(19,31)(20,35)(21,27)(22,30)(23,29)(24,28)(25,34)(26,36)(32,44)(33,42)(37,41)(38,43)(39,45)(40,47)(46,48);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,32)(21,37)(22,39)(23,34)(24,33)(26,40)(27,44)(28,45)(29,31)(30,42)(35,41)(36,48)(38,47)(43,46);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,29,43)(19,32,34,37)(22,36,24,38)(25,40,31,46)(27,28,35,30)(33,47,39,48)(41,42,44,45);(1,3,5,4)(2,6,7,8)(10,14)(11,16)(17,21,23,20)(18,27,29,35)(19,32,34,37)(22,36,24,38)(25,41,31,44)(26,28,43,30)(33,48,39,47)(40,42,46,45);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,33,34,39)(20,36,21,38)(25,42,31,45)(26,35,43,27)(32,47,37,48)(40,44,46,41)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=f,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=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, 7214820);