#7779432

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	#50599	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=gi,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, 7686628);