#9947453

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=gh,f^2=g,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, 9854649);