#478197

order	512
nilpotency class	3
isoclinism	isoclinic class of #478195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=efi,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385393);