#8932711

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839907);