#7304567

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fi,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, 7211763);