#9685549

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3746 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9592745);