#8505937

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413133);