#8445888

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18,24,22)(19,29,31,25)(20,27,21,26)(23,28,30,32)(33,34,39,43)(35,44,45,40)(36,41,46,48)(37,38,42,47);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,19)(18,25)(20,30)(21,23)(22,29)(24,31)(26,32)(27,28)(35,45)(36,46)(40,44)(41,48);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,20,24,21)(18,26,22,27)(19,30,31,23)(25,32,29,28)(33,35)(34,40)(36,42)(37,46)(38,41)(39,45)(43,44)(47,48);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,21,24,20)(18,27,22,26)(19,30,31,23)(25,32,29,28)(33,36,39,46)(34,41,43,48)(35,42,45,37)(38,44,47,40);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22,24,18)(19,29,31,25)(20,27,21,26)(23,32,30,28)(33,37,39,42)(34,38,43,47)(35,46,45,36)(40,41,44,48);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,23,24,30)(18,28,22,32)(19,21,31,20)(25,27,29,26)(33,38)(34,42)(35,41)(36,40)(37,43)(39,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=h,e^2=gi,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, 8353084);