#401024

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1744 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308220);