#8755247

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	#57469	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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)(3,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34,39,42)(35,43,46,37)(36,41,38,40)(44,47,45,48);(1,2,7,4)(3,8,5,11)(6,12,14,9)(10,16,13,15)(21,29)(24,28)(26,32)(30,31)(33,35,39,46)(34,37,42,43)(36,44,38,45)(40,47,41,48);(1,3,7,5)(2,8,4,11)(6,10,14,13)(9,15,12,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35)(34,37)(36,44)(38,45)(39,46)(40,47)(41,48)(42,43);(1,4,7,2)(3,8,5,11)(6,12,14,9)(10,15,13,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36,39,38)(34,40,42,41)(35,44,46,45)(37,47,43,48);(1,5,7,3)(2,8,4,11)(6,13,14,10)(9,15,12,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37)(34,35)(36,47)(38,48)(39,43)(40,44)(41,45)(42,46);(1,6,7,14)(2,9,4,12)(3,10,5,13)(8,15,11,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,38,39,36)(34,41,42,40)(35,45,46,44)(37,48,43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=ghi,e^2=gh,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, 8662443);