#8463749

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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:

(2,6)(3,9)(8,11)(10,15)(18,20)(23,26)(24,28)(30,32)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(19,25)(23,26)(27,31)(30,32)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(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=i,c^2=h,e^2=ghi,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, 8370945);