#430953

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1111 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338149);