#7777928

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	#41955	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4580 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=ghi,e^2=i,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, 7685124);