#8094539

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3858 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001735);