#8359688

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

4113 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8266884);