#9715497

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9622693);