#7304571

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,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, 7211767);