#9783093

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,22,24)(19,26,29,23)(20,21,30,25)(27,31,32,28)(33,34,39,41)(35,40,43,38)(36,37,42,48)(44,45,47,46);(4,11)(8,14)(10,15)(12,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,31)(30,32)(33,35)(34,40)(36,44)(37,45)(38,41)(39,43)(42,47)(46,48);(2,7)(3,9)(8,14)(10,15)(18,24)(20,30)(23,26)(27,32)(34,41)(35,43)(36,42)(45,46);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20)(18,25)(19,27)(21,24)(22,30)(23,31)(26,28)(29,32)(33,36,39,42)(34,37,41,48)(35,44,43,47)(38,46,40,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,19)(18,26)(20,27)(21,31)(22,29)(23,24)(25,28)(30,32)(33,37)(34,42)(35,45)(36,41)(38,44)(39,48)(40,47)(43,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,25)(18,20,24,30)(19,28,29,31)(23,27,26,32)(33,38)(34,43)(35,41)(36,46)(37,47)(39,40)(42,45)(44,48)

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,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, 9690289);