#8095218

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	#45592	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^2=gi,f^2=h,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, 8002414);