#9916493

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(35,42)(40,46)(43,47)(44,48);(1,3,7,4)(2,8,10,5)(6,11,15,12)(9,14,16,13)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(34,39)(36,41)(40,46)(43,47);(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,34,38,39)(35,40,42,46)(36,37,41,45)(43,44,47,48);(1,4,7,3)(2,8,10,5)(6,12,15,11)(9,14,16,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,40)(36,43)(37,44)(38,42)(39,46)(41,47)(45,48);(1,5,7,8)(2,3,10,4)(6,14,15,13)(9,12,16,11)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36)(34,37)(35,43)(38,41)(39,45)(40,44)(42,47)(46,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,14)(10,16)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31)(33,37)(34,41)(35,44)(36,39)(38,45)(40,47)(42,48)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219279, #72219283
maximal quotient of	#72219270

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9823689);