#8508933

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	#39308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,23,28)(19,25,20,24)(21,29,31,26)(22,27,32,30)(33,34)(35,44)(36,42)(37,41)(38,40)(39,43)(45,47)(46,48);(3,4)(5,13)(7,10)(8,15)(17,19,23,20)(18,24,28,25)(21,30,31,27)(22,29,32,26)(33,35,39,38)(34,40,43,44)(36,45,37,46)(41,47,42,48);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,19,23,20)(18,25,28,24)(21,27,31,30)(22,29,32,26)(33,36,39,37)(34,41,43,42)(35,45,38,46)(40,47,44,48);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,20,23,19)(18,25,28,24)(21,30,31,27)(22,29,32,26)(33,37,39,36)(34,42,43,41)(35,45,38,46)(40,47,44,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,21,23,31)(18,26,28,29)(19,27,20,30)(22,24,32,25)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,48)(46,47);(1,2)(3,7)(4,10)(5,8)(6,9)(11,14)(12,16)(13,15)(17,22)(18,27)(19,29)(20,26)(21,24)(23,32)(25,31)(28,30)(33,38,39,35)(34,40,43,44)(36,46,37,45)(41,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=gi,e^2=ghi,f^2=hi,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, 8416129);