#9815118

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

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, 9722314);