#324121

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eghi,c^2=fi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231317);