#7409924

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1605 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)(17,18,22,26)(19,27,29,40)(20,24,31,38)(21,33,35,25)(23,34,37,30)(28,41,42,48)(32,45,44,39)(36,46,47,43);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(41,48)(43,46);(10,12)(11,15)(17,19,22,29)(18,23,26,37)(20,28,31,42)(21,34,35,30)(24,36,38,47)(25,40,33,27)(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)(18,24)(19,28)(21,32)(22,31)(23,36)(25,39)(26,38)(27,41)(29,42)(30,43)(33,45)(34,46)(35,44)(37,47)(40,48);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(19,29)(20,31)(23,37)(24,38)(27,40)(30,34)(32,44)(39,45)

Polycyclic presentation:

2^9|a^2=fhi,c^2=i,d^2=fh,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=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, 7317120);