#8931101

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(3,4)(5,13)(6,8)(7,11)(9,14)(10,15)(12,16)(17,18,22,26)(19,25,29,23)(20,24,30,21)(27,28,31,32)(35,41)(39,46)(42,44)(47,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,13)(14,16)(17,19,22,29)(18,23,26,25)(20,27,30,31)(21,28,24,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,8)(4,7)(5,14)(10,16)(20,30)(21,24)(27,31)(28,32)(34,38)(35,41)(40,43)(42,44);(1,4,6,7)(2,3,8,11)(5,12,14,15)(9,10,13,16)(17,20)(18,24)(19,27)(21,26)(22,30)(23,32)(25,28)(29,31)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,15)(14,16)(17,19)(18,25)(20,31)(21,28)(22,29)(23,26)(24,32)(27,30)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48);(1,5,6,14)(2,9,8,13)(3,10,11,16)(4,12,7,15)(17,21,22,24)(18,20,26,30)(19,28,29,32)(23,27,25,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=i,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, 8838297);