#402604

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,d^2=e,f^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,d^c=di,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, 309800);