#6630069

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ci,e^c=ei,e^d=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, 6537265);