#8930335

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

4327 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=ghi,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, 8837531);