#7409911

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=i,d^2=fhi,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,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, 7317107);