#9715495

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

3911 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=g,f^2=gh,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, 9622691);