#7308785

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7215981);