#8762043

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	#48458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,e^2=g,f^2=g,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, 8669239);