#9453092

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	#37022	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3906 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)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34)(35,41)(36,44)(37,43)(38,39)(40,45)(42,47)(46,48);(4,11)(7,9)(12,16)(14,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,38,43)(34,37,39,41)(36,42,45,46)(40,47,44,48);(2,3)(7,9)(8,10)(14,15)(18,19)(21,29)(23,25)(27,32)(34,39)(36,45)(37,41)(42,46);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,36)(34,40)(35,42)(37,47)(38,45)(39,44)(41,48)(43,46);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,35)(34,41)(36,46)(37,39)(38,43)(40,47)(42,45)(44,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,41)(34,35,39,43)(36,47,45,48)(40,42,44,46)

Polycyclic presentation:

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

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, 9360288);