#8927132

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

4088 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)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(33,34,39,37)(35,38,44,40)(36,42,46,41)(43,45,48,47);(2,7)(4,13)(8,15)(10,14)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(33,35,39,44)(34,40,37,38)(36,43,46,48)(41,45,42,47);(3,11)(5,16)(8,15)(10,14)(19,27)(20,28)(22,32)(24,29)(33,36,39,46)(34,41,37,42)(35,43,44,48)(38,47,40,45);(1,3)(2,8)(4,14)(5,12)(6,11)(7,15)(9,16)(10,13)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,36)(34,42)(35,43)(37,41)(38,45)(39,46)(40,47)(44,48);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,15,16,8)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,37,39,34)(35,38,44,40)(36,42,46,41)(43,47,48,45);(1,5)(2,10)(3,12)(4,15)(6,16)(7,14)(8,13)(9,11)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,38)(34,35)(36,45)(37,44)(39,40)(41,48)(42,43)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8834328);