#8772796

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	#53172	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679992);