#8926653

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833849);