#8095164

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

3845 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)(17,18,23,28)(19,21,26,24)(20,27,22,25)(29,32,30,31)(33,34)(35,43)(36,38)(37,42)(39,40)(41,47)(44,45)(46,48);(2,6)(7,9)(8,13)(14,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,26)(25,32)(27,31)(34,40)(36,42)(41,43)(44,48);(1,2,5,6)(3,9,11,7)(4,8,12,13)(10,15,16,14)(17,19)(18,21)(20,29)(22,30)(23,26)(24,28)(25,31)(27,32)(33,35,39,47)(34,41,40,43)(36,44,42,48)(37,45,38,46);(17,20,23,22)(18,25,28,27)(19,29,26,30)(21,31,24,32)(33,36)(34,37)(35,44)(38,40)(39,42)(41,45)(43,46)(47,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21)(18,26)(19,28)(20,31)(22,32)(23,24)(25,30)(27,29)(33,37,39,38)(34,42,40,36)(35,45,47,46)(41,48,43,44);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,20)(18,27,28,25)(19,30,26,29)(21,32,24,31)(33,38,39,37)(34,36,40,42)(35,46,47,45)(41,44,43,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=gh,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, 8002360);