#9938024

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	#54841	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9845220);