#10247132

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

4050 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154328);