#8932756

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
prime radical	#46472	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=g,f^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, 8839952);