#9812896

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	#54922	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9720092);