#8093254

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	#45741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,e^2=g,f^2=h,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, 8000450);