#8095033

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,e^2=i,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, 8002229);