#8755408

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662604);