#8924728

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	#57992	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=hi,e^2=gi,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, 8831924);