#9062099

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

4020 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969295);