#9910006

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	#49838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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)(3,4,8,11)(5,9,14,15)(10,12,16,13)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(2,7)(4,11)(9,15)(12,13)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(18,19)(23,25)(24,26)(30,31)(34,40)(36,44)(39,46)(41,47);(1,3,6,8)(2,4,7,11)(5,10,14,16)(9,12,15,13)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,4)(2,8)(3,7)(5,13)(6,11)(9,10)(12,14)(15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817202);