#8752012

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	#45638	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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)(7,10)(11,15)(14,16)(17,18,23,24)(19,21,27,30)(20,25,28,32)(22,29,31,26)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19)(18,21)(20,22)(23,27)(24,30)(25,29)(26,32)(28,31)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(2,6)(7,10)(8,12)(14,16)(18,24)(19,27)(20,28)(26,29)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,28)(18,25,24,32)(19,22,27,31)(21,29,30,26)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21)(18,19)(20,29)(22,25)(23,30)(24,27)(26,28)(31,32)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,26)(19,28)(20,27)(21,25)(23,31)(24,29)(30,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^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, 8659208);