#8930170

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	#55052	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837366);