#7771958

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4267 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7679154);