#8445423

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

4225 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gh,e^2=hi,f^2=h,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, 8352619);