#7726692

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7633888);