#7485227

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=f,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=efi,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, 7392423);