#8492766

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	#52375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8399962);