#7304565

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

1353 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)(25,26,31,36)(27,32,41,50)(28,42,44,34)(29,35,30,46)(33,45,51,43)(37,48,38,52)(39,49,40,55)(47,54,56,53);(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)(28,37)(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)(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)(27,37)(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)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(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);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(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=i,b^2=f,c^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7211761);