#8324082

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=h,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, 8231278);