#9018743

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	#44318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(33,34,39,43)(35,38,37,41)(36,42,47,40)(44,46,45,48);(2,7)(4,14)(8,11)(10,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35,39,37)(34,38,43,41)(36,45,47,44)(40,46,42,48);(2,7)(3,12)(4,14)(5,16)(18,22)(19,25)(20,28)(30,32)(33,36,39,47)(34,40,43,42)(35,44,37,45)(38,48,41,46);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,37,39,35)(34,41,43,38)(36,44,47,45)(40,48,42,46);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36)(34,42)(35,44)(37,45)(38,46)(39,47)(40,43)(41,48);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,38,39,41)(34,35,43,37)(36,46,47,48)(40,45,42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8925939);