#8932729

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	#46519	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=g,f^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, 8839925);