#8174292

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

4193 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)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(34,36)(38,46)(40,41)(43,45);(2,7)(3,10)(8,14)(11,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081488);