#7597467

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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)(11,15)(12,13)(14,16)(17,18,22,26)(19,27,31,42)(20,32,34,46)(21,35,29,25)(23,36,38,28)(24,39,40,33)(30,44,45,37)(41,47,48,43);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(19,28)(20,33)(21,29)(23,27)(24,32)(25,35)(30,45)(31,36)(34,39)(37,44)(38,42)(40,46);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,29,36,22,31,21,28)(18,23,35,42,26,38,25,27)(20,30,39,47,34,45,33,43)(24,37,46,48,40,44,32,41);(9,11,12,16)(10,14,13,15)(17,20,29,39,22,34,21,33)(18,24,35,46,26,40,25,32)(19,30,36,47,31,45,28,43)(23,37,42,48,38,44,27,41);(1,4,5,3)(2,7,6,8)(19,31)(23,38)(27,42)(28,36)(30,45)(37,44)(41,48)(43,47)

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=gi,e^2=f,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^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, 7504663);