#8095529

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	#45610	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,24)(20,26)(21,29)(22,23)(25,30)(27,28)(31,32)(37,45)(42,46)(43,48)(44,47);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,8)(4,9)(7,14)(10,15)(18,23)(19,26)(25,29)(27,32)(34,39)(36,40)(42,46)(44,47);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,47,46,44);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,21,22,30)(18,25,23,29)(19,27,26,32)(20,28,24,31)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|b^2=gh,d^2=i,e^2=hi,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, 8002725);