#9481470

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	#54045	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3767 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,24,27)(19,28,21,25)(20,23,22,26)(29,32,30,31)(33,34)(35,43)(36,37)(38,40)(39,42)(41,48)(44,45)(46,47);(3,9)(7,13)(10,15)(14,16)(17,19,24,21)(18,25,27,28)(20,29,22,30)(23,31,26,32)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(2,6)(7,13)(8,11)(14,16)(17,20,24,22)(18,26,27,23)(19,29,21,30)(25,32,28,31)(33,36,39,41)(34,37,42,48)(35,44,38,46)(40,47,43,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,21,24,19)(18,25,27,28)(20,29,22,30)(23,32,26,31)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,22,24,20)(18,26,27,23)(19,30,21,29)(25,32,28,31);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,23,24,26)(18,20,27,22)(19,31,21,32)(25,29,28,30)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388666);