#8493089

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4202 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400285);