#8930198

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

4137 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)(3,8,11,15)(4,9,13,12)(5,10,16,14)(19,25)(23,30)(26,28)(31,32)(33,34,39,43)(35,40,42,37)(36,45,38,41)(44,48,46,47);(2,7)(4,13)(8,15)(10,14)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,39,42)(34,40,43,37)(36,46,38,44)(41,48,45,47);(3,11)(5,16)(8,15)(10,14)(18,22)(19,25)(24,27)(26,28)(33,36,39,38)(34,41,43,45)(35,44,42,46)(37,48,40,47);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,37,39,40)(34,42,43,35)(36,47,38,48)(41,44,45,46);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,38,39,36)(34,41,43,45)(35,44,42,46)(37,47,40,48);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,34)(35,40)(36,45)(37,42)(38,41)(39,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gi,e^2=h,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, 8837394);