#9814928

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gi,f^2=h,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, 9722124);