#8324400

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

4054 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,e^2=h,f^2=hi,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, 8231596);