#7603271

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1593 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)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18,22,26)(19,28,30,24)(20,31,33,43)(21,34,36,48)(23,38,35,27)(25,40,42,37)(29,45,46,41)(32,44,47,39);(2,6)(7,8)(10,13)(11,16)(17,19,22,30)(18,24,26,28)(20,23,33,35)(21,32,36,47)(25,41,42,45)(27,43,38,31)(29,40,46,37)(34,39,48,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,23,30,22,33,35,19)(18,24,27,31,26,28,38,43)(21,32,37,46,36,47,40,29)(25,39,34,41,42,44,48,45);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,35,40,22,36,23,37)(18,25,38,48,26,42,27,34)(19,29,33,47,30,46,20,32)(24,39,43,45,28,44,31,41);(2,6)(7,8)(10,13)(14,15)(18,26)(21,36)(24,28)(27,38)(29,46)(31,43)(32,47)(37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fhi,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7510467);