#8757439

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	#45986	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(3,11)(4,13)(5,15)(6,7)(8,12)(9,14)(10,16)(17,18)(19,23)(20,26)(21,24)(22,30)(25,28)(27,29)(31,32)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(2,7)(5,16)(8,11)(9,13)(17,19)(18,25)(20,27)(21,23)(22,31)(24,28)(26,32)(29,30)(34,37)(35,42)(41,45)(43,48);(2,7)(3,12)(9,13)(10,15)(17,20,24,22)(18,26,21,30)(19,27,28,31)(23,29,25,32)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,16)(14,15)(17,21,24,18)(19,23,28,25)(20,26,22,30)(27,32,31,29)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,22,24,20)(18,26,21,30)(19,27,28,31)(23,32,25,29)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,23)(18,19)(20,29)(21,28)(22,32)(24,25)(26,27)(30,31)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,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, 8664635);