#8095521

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	#53061	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(37,45)(42,46)(43,48)(44,47);(3,9)(7,10)(11,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,4)(7,10)(8,12)(14,16)(19,25)(22,29)(26,31)(30,32)(34,39)(36,40)(42,46)(44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,47,46,44);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gh,d^2=i,e^2=ghi,f^2=g,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, 8002717);