#8115237

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022433);