#7228819

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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

1360 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,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, 7136015);