#8446102

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	#52943	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=g,e^2=hi,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, 8353298);