#7485192

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1280 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)(25,26,31,33,30,36,29,35)(27,32,44,46,39,49,38,48)(28,40,45,52,42,34,43,50)(37,51,54,56,41,47,53,55);(1,2,4,5)(3,7,8,6)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,39)(26,32,36,49)(28,41,42,37)(29,38,31,44)(33,46,35,48)(34,51,40,47)(43,54,45,53)(50,56,52,55);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(29,31)(32,46)(34,50)(35,36)(38,44)(40,52)(43,45)(47,55)(48,49)(51,56)(53,54);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,37)(29,43)(30,42)(31,45)(32,47)(33,50)(35,52)(36,40)(38,53)(39,41)(44,54)(46,55)(48,56)(49,51);(1,3,4,8)(2,6,5,7)(10,18)(15,19)(16,17)(23,24)(28,42)(34,40)(37,41)(43,45)(47,51)(50,52)(53,54)(55,56)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=f,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,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, 7392388);