#8438030

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	#41445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4140 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345226);