#8475178

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

4133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=gh,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, 8382374);