#9863942

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3807 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)(3,7)(4,8)(5,13)(6,10)(9,15)(11,12)(14,16)(17,18,23,21)(19,27,30,24)(20,31,29,25)(22,26,28,32);(1,3,6,8)(2,7,10,4)(5,11,15,16)(9,14,13,12)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,32)(26,31)(33,34,39,36)(35,42,40,37)(38,41,47,44)(43,46,48,45);(1,2,6,10)(3,4,8,7)(5,9,15,13)(11,12,16,14)(17,18,23,21)(19,24,30,27)(20,25,29,31)(22,26,28,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,16)(13,15)(17,20,23,29)(18,25,21,31)(19,28,30,22)(24,32,27,26)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,4,6,7)(2,8,10,3)(5,14,15,12)(9,11,13,16)(17,21,23,18)(19,24,30,27)(20,31,29,25)(22,26,28,32)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,5)(2,9)(3,11)(4,12)(6,15)(7,14)(8,16)(10,13)(17,22,23,28)(18,26,21,32)(19,29,30,20)(24,31,27,25)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9771138);