#7337874

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,35)(27,31,38,34)(28,39,41,43)(29,33,42,48)(32,45,47,36)(37,44,51,49)(40,52,53,54)(46,55,56,50);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,35,34)(28,36,41,45)(29,37,42,51)(32,43,47,39)(33,44,48,49)(40,50,53,55)(46,54,56,52);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,38,45,30,41,27,36)(26,32,31,43,35,47,34,39)(29,40,51,55,42,53,37,50)(33,46,44,54,48,56,49,52);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,29,30,42)(26,33,35,48)(27,37,38,51)(28,40,41,53)(31,44,34,49)(32,46,47,56)(36,50,45,55)(39,52,43,54);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,41)(29,42)(32,47)(33,48)(36,45)(37,51)(39,43)(44,49)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=fi,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=eh,d^c=dh,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, 7245070);