#7409021

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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)(19,27)(20,25)(21,23)(22,35)(24,30)(26,34)(28,42)(29,43)(31,38)(32,46)(33,48)(36,37)(39,44)(40,47)(41,45);(1,3)(2,6)(4,8)(5,7)(9,10,13,15)(11,14,12,16)(18,23)(19,28)(20,31)(26,35)(30,36)(33,45)(39,40)(43,46);(2,4)(3,7)(9,10)(11,16)(12,14)(13,15)(17,19,21,28)(18,24,23,37)(20,32,31,29)(22,36,34,30)(25,40,38,39)(26,42,35,27)(33,44,45,47)(41,43,48,46);(1,4,5,2)(3,6,7,8)(9,11,13,12)(10,14,15,16)(17,20,21,31)(18,25,23,38)(19,29,28,32)(22,33,34,45)(24,39,37,40)(26,41,35,48)(27,43,42,46)(30,44,36,47);(3,7)(6,8)(9,12,13,11)(10,14,15,16)(18,23)(19,28)(20,31)(26,35)(30,36)(33,45)(39,40)(43,46)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=h,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=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, 7316217);