#7037505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1237 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)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,37)(27,39,43,33)(28,44,29,35)(30,38,32,36)(34,46,52,45)(40,49,41,53)(42,50,47,54)(48,56,51,55);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,43)(26,33,37,39)(28,41,29,40)(30,47,32,42)(34,51,52,48)(35,53,44,49)(36,54,38,50)(45,55,46,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,31,29)(26,34,37,52)(27,40,43,41)(30,45,32,46)(33,48,39,51)(35,36,44,38)(42,55,47,56)(49,54,53,50);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,29,31,28)(26,35,37,44)(27,41,43,40)(30,45,32,46)(33,49,39,53)(34,36,52,38)(42,55,47,56)(48,54,51,50);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,30,31,32)(26,36,37,38)(27,42,43,47)(28,45,29,46)(33,50,39,54)(34,44,52,35)(40,55,41,56)(48,49,51,53)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6944701);