#8926315

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	#54199	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=h,e^2=g,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, 8833511);