#7489374

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1368 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)(25,26,29,34)(27,36,37,31)(28,40,42,33)(30,32,44,35)(38,46,41,49)(39,45,51,47)(43,50,54,48)(52,56,53,55);(2,6)(3,4)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,29,37)(26,31,34,36)(28,41,42,38)(30,39,44,51)(32,47,35,45)(33,49,40,46)(43,53,54,52)(48,55,50,56);(1,2)(3,8)(4,7)(5,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,37)(28,42)(30,44)(31,45)(33,50)(34,35)(36,47)(40,48)(46,56)(49,55)(52,53);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,42)(26,33,34,40)(27,38,37,41)(30,43,44,54)(31,46,36,49)(32,48,35,50)(39,52,51,53)(45,55,47,56);(1,4,5,3)(2,7,6,8)(10,18)(15,19)(16,17)(23,24)(26,34)(27,37)(28,42)(32,35)(39,51)(43,54)(46,49)(55,56)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fhi,e^2=hi,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, 7396570);