#8192552

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

4126 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8099748);