#8324117

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

3914 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231313);