#9863396

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

4094 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,21,23)(19,25,27,22)(20,24,29,31)(26,32,28,30)(33,34,39,44)(35,40,46,37)(36,42,48,41)(38,47,45,43);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32)(33,35,39,46)(34,40,44,37)(36,45,48,38)(41,47,42,43);(2,4)(3,10)(9,12)(11,16)(18,23)(22,25)(24,31)(30,32)(33,36,39,48)(34,41,44,42)(35,38,46,45)(37,47,40,43);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)(33,36)(34,42)(35,45)(37,47)(38,46)(39,48)(40,43)(41,44);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32)(33,37)(34,35)(36,47)(38,41)(39,40)(42,45)(43,48)(44,46);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,38,39,45)(34,43,44,47)(35,36,46,48)(37,42,40,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=ghi,e^2=hi,f^2=gh,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, 9770592);