#7601047

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,29,33,40)(20,34,37,25)(21,27,22,26)(24,31,41,32)(30,46,35,45)(36,43,38,44)(39,47,42,48);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,31,23,33,22,32)(18,24,26,40,28,41,27,29)(20,35,38,47,37,30,36,48)(25,42,44,45,34,39,43,46);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(17,20)(18,25)(19,30)(21,38)(22,36)(23,37)(24,39)(26,44)(27,43)(28,34)(29,45)(31,48)(32,47)(33,35)(40,46)(41,42);(2,5)(6,7)(10,14)(11,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,40,41,29)(25,43,34,44)(30,47,35,48)(39,45,42,46);(2,5)(6,7)(10,14)(13,15)(17,19,21,31,23,33,22,32)(18,24,26,40,28,41,27,29)(20,35,38,47,37,30,36,48)(25,42,44,45,34,39,43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,d^2=i,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7508243);