#8821836

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8729032);