#8475102

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382298);