#7409151

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=fhi,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=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7316347);