#413024

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
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

674 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)(7,18,21,31,23,33,22,13)(8,24,9,28,26,14,27,32)(19,30,35,34,20,36,25,29);(1,2)(3,4)(5,7,12,21,11,23,10,22)(6,13,17,18,16,31,15,33)(8,25,27,19,26,35,9,20)(14,30,28,34,24,36,32,29);(2,4)(5,8,10,9,11,26,12,27)(6,14,15,32,16,24,17,28)(7,19,21,35,23,20,22,25)(13,29,18,30,31,34,33,36);(5,9,10,26,11,27,12,8)(6,14,17,28,16,24,15,32)(7,20,22,35,23,19,21,25)(13,30,18,34,31,36,33,29)

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,d^c=di

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, 320220);