#8932762

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

4171 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)(3,8)(4,12)(5,13)(6,7)(9,15)(10,16)(11,14)(19,25)(20,28)(23,26)(24,29)(33,34,39,42)(35,36,40,37)(38,41,46,48)(43,44,47,45);(2,7)(3,11)(9,12)(10,13)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,35,39,40)(34,37,42,36)(38,43,46,47)(41,45,48,44);(2,7)(5,16)(8,14)(9,12)(18,22)(23,26)(24,29)(31,32)(33,36,39,37)(34,40,42,35)(38,45,46,44)(41,43,48,47);(1,3)(2,8)(4,13)(5,12)(6,11)(7,14)(9,16)(10,15)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,4,6,15)(2,9,7,12)(3,10,11,13)(5,14,16,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,37,39,36)(34,40,42,35)(38,45,46,44)(41,47,48,43);(1,5)(2,10)(3,12)(4,14)(6,16)(7,13)(8,15)(9,11)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839958);