#10363700

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10270896);