#8930230

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

3975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=hi,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, 8837426);