#8295572

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	#56809	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

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, 8202768);