#7409036

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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,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, 7316232);