#405788

order	512
nilpotency class	3
isoclinism	isoclinic class of #395409
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 312984);