#7596318

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

1538 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,29)(19,24,33,42)(20,34,39,45)(21,27,22,28)(25,31,41,32)(26,38,46,37)(30,44,36,43)(35,48,47,40);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,28)(26,43)(27,41)(29,42)(30,38)(34,48)(36,37)(39,47)(40,45)(44,46);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,19,22,31,23,33,21,32)(18,25,27,42,29,41,28,24)(20,36,38,47,39,30,37,35)(26,44,45,40,46,43,34,48);(1,4,5,3)(2,6,8,7)(9,11,12,16)(10,14,13,15)(17,20,21,37,23,39,22,38)(18,26,28,34,29,46,27,45)(19,30,32,47,33,36,31,35)(24,40,41,44,42,48,25,43);(2,8)(6,7)(17,21,23,22)(18,27,29,28)(19,31,33,32)(20,37,39,38)(24,41,42,25)(26,45,46,34)(30,35,36,47)(40,44,48,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,d^2=fg,e^2=i,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,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, 7503514);