#7590440

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1357 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)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,27,29,41)(20,24,32,38)(21,34,30,25)(23,35,37,31)(28,40,44,48)(33,45,42,39)(36,46,47,43);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,31,22,29,30,35)(18,23,25,27,26,37,34,41)(20,28,33,43,32,44,42,46)(24,36,39,40,38,47,45,48);(1,3,4,8)(2,5,6,7)(17,20,22,32)(18,24,26,38)(19,28,29,44)(21,33,30,42)(23,36,37,47)(25,39,34,45)(27,40,41,48)(31,43,35,46);(2,6)(5,7)(11,15)(13,16)(19,29)(20,32)(23,37)(24,38)(27,41)(31,35)(33,42)(39,45);(17,19,30,35,22,29,21,31)(18,23,34,41,26,37,25,27)(20,28,42,46,32,44,33,43)(24,36,45,48,38,47,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=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, 7497636);