#8930941

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3776 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,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, 8838137);