#8425648

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	#53046	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332844);