#8031598

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	#45404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

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, 7938794);