#413023

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

684 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=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, 320219);