#8925702

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	#54834	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=gh,f^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, 8832898);