#9267032

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	#56351	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3856 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9174228);