#8183739

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	#53063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4135 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090935);