#8492912

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

4016 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18,23,28)(19,25,20,24)(21,22,27,26)(29,30,31,32)(33,34,38,39)(35,36,41,40)(37,42,46,45)(43,44,48,47);(2,7)(3,10)(8,14)(11,16)(17,19,23,20)(18,24,28,25)(21,31,27,29)(22,30,26,32)(34,39)(36,40)(42,45)(44,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,23,20)(18,25,28,24)(21,29,27,31)(22,30,26,32)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20,23,19)(18,25,28,24)(21,31,27,29)(22,30,26,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,44)(47,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21,23,27)(18,26,28,22)(19,29,20,31)(24,30,25,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,22,23,26)(18,27,28,21)(19,30,20,32)(24,29,25,31)(33,37,38,46)(34,42,39,45)(35,43,41,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8400108);