#8467039

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	#53103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4190 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=h,e^2=hi,f^2=gi,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, 8374235);