#7771981

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	#41165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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)(9,10,14,16)(11,17,18,15)(12,21,23,13)(19,20,22,24);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,20)(14,18)(16,17)(21,24)(22,23)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,5,4)(2,7,6,8)(10,16)(11,18)(13,21)(19,22)(26,31)(29,38)(32,39)(36,40);(9,11)(10,17)(12,22)(13,20)(14,18)(15,16)(19,23)(21,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(9,12)(10,13)(11,19)(14,23)(15,20)(16,21)(17,24)(18,22)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,35)(38,40);(1,4,5,3)(2,8,6,7)(9,13)(10,12)(11,20)(14,21)(15,19)(16,23)(17,22)(18,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=hi,f^2=gh,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, 7679177);