#412737

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,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, 319933);