#7711638

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,30,35)(27,36,38,31)(28,33,39,32)(29,40,37,34);(1,2)(3,6)(4,5)(7,8)(25,27,30,38)(26,31,35,36)(28,37,39,29)(32,34,33,40);(10,11)(15,17)(16,18)(22,23)(25,28,30,39)(26,32,35,33)(27,29,38,37)(31,40,36,34);(1,3,4,7)(2,6,5,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,33)(27,37)(29,38)(30,39)(31,40)(32,35)(34,36);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,31)(28,33)(29,34)(30,35)(32,39)(36,38)(37,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,37)(26,34,35,40)(27,28,38,39)(31,33,36,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=gh,e^2=hi,f^2=h,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, 7618834);