#414692

order	512
nilpotency class	3
isoclinism	isoclinic class of #394040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321888);