#8928015

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

4192 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gh,e^2=gh,f^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, 8835211);