#7799527

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	#39881	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4535 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)(11,18)(12,16)(13,15)(14,17)(19,24)(20,22)(21,23)(25,26,31,30)(27,35,29,32)(28,34,38,33)(36,39,37,40);(1,2,4,5)(3,6,7,8)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,27)(26,32)(28,37)(29,31)(30,35)(33,40)(34,39)(36,38);(11,13)(15,18)(19,21)(23,24)(25,28,31,38)(26,33,30,34)(27,36,29,37)(32,39,35,40);(1,3,4,7)(2,6,5,8)(9,12,14,22)(10,16,17,20)(11,19,13,21)(15,23,18,24)(25,28)(26,34)(27,37)(29,36)(30,33)(31,38)(32,39)(35,40);(9,10)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,29,31,27)(26,35,30,32)(28,37,38,36)(33,40,34,39);(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,30,31,26)(27,35,29,32)(28,33,38,34)(36,39,37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7706723);