#7204040

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,35)(27,32,30,45)(28,38,41,46)(29,42,43,34)(33,39,49,36)(37,51,44,47)(40,54,55,52)(48,50,56,53);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,30)(26,32,35,45)(28,39,41,36)(29,37,43,44)(33,38,49,46)(34,47,42,51)(40,53,55,50)(48,52,56,54);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,36)(29,40)(30,39)(31,41)(32,46)(34,48)(35,49)(37,50)(38,45)(42,56)(43,55)(44,53)(47,54)(51,52);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,31,43)(26,34,35,42)(27,37,30,44)(28,40,41,55)(32,47,45,51)(33,48,49,56)(36,50,39,53)(38,52,46,54);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(28,41)(29,43)(33,49)(34,42)(36,39)(37,44)(38,46)(47,51)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,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, 7111236);