#7304611

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1344 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,36)(27,37,41,32)(28,34,44,50)(29,35,31,46)(33,45,51,43)(38,48,39,52)(40,54,42,49)(47,55,53,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,38)(29,40)(30,41)(31,42)(33,47)(34,52)(35,49)(36,37)(39,44)(43,55)(45,56)(46,54)(48,50)(51,53);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,38)(29,43)(30,44)(31,45)(32,47)(34,46)(35,50)(36,51)(37,53)(39,41)(40,55)(42,56)(48,49)(52,54);(2,6)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,39)(29,45)(30,44)(31,43)(32,48)(33,46)(35,51)(36,50)(37,52)(38,41)(40,55)(42,56)(47,49)(53,54);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,35,36,46)(27,40,41,42)(28,43,44,45)(32,49,37,54)(33,50,51,34)(38,55,39,56)(47,48,53,52)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7211807);