#7712682

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4500 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)(12,20)(16,21)(18,24)(22,23)(25,26,30,34)(27,31,38,35)(28,39,37,32)(29,40,36,33);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,21)(22,23)(25,26,30,34)(27,31,38,35)(28,32,37,39)(29,33,36,40);(1,3,4,7)(2,6,5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,35)(28,39)(29,33)(30,34)(31,38)(32,37)(36,40);(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,37)(26,32,34,39)(27,36,38,29)(31,40,35,33);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619878);