#8322400

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3860 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8229596);