#8922634

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	#58904	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829830);