#7715548

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	#37559	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 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)(11,19)(12,17)(13,15)(14,24)(16,20)(18,21)(22,23)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,14)(13,19)(15,20)(17,18)(21,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(10,13)(11,20)(12,22)(18,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(9,12)(10,17)(11,21)(13,23)(14,20)(15,22)(16,24)(18,19)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(9,13,15,10)(11,19,20,16)(12,23,22,17)(14,24,21,18)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(9,14,15,21)(10,18,13,24)(11,12,20,22)(16,17,19,23)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622744);