#8460704

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(18,24)(19,20)(26,30)(27,29)(33,34,38,40)(35,41,42,36)(37,39,46,48)(43,45,47,44);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(17,18,23,24)(19,25,20,28)(21,26,31,30)(22,27,32,29)(35,42)(36,41)(43,47)(44,45);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,26,32,30)(33,35)(34,36)(37,43)(38,42)(39,44)(40,41)(45,48)(46,47);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,20,23,19)(18,25,24,28)(21,29,31,27)(22,26,32,30)(33,36)(34,35)(37,45)(38,41)(39,47)(40,42)(43,48)(44,46);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,21,23,31)(18,26,24,30)(19,27,20,29)(22,25,32,28);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,22,23,32)(18,27,24,29)(19,26,20,30)(21,25,31,28)(33,37)(34,39)(35,43)(36,44)(38,46)(40,48)(41,45)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=hi,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, 8367900);