#8227500

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8134696);