#412432

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1358 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fi,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, 319628);