#8434134

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	#45556	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341330);