#8093197

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	#46032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(6,13)(10,15)(14,16)(20,27)(24,29)(26,31)(30,32)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(18,22)(19,25)(24,29)(26,31)(34,39)(37,40)(41,44)(45,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000393);