#7830797

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	#59009	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4503 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,e^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7737993);