#434068

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1149 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)(5,6,12,16,11,18,10,17)(7,19,22,34,23,33,21,13)(8,24,25,35,27,29,26,14)(9,28,32,15,31,36,30,20);(5,7,10,21,11,23,12,22)(6,13,16,19,18,34,17,33)(8,20,25,28,27,15,26,36)(9,29,30,35,31,24,32,14);(5,8,11,27)(6,14)(7,20)(9,13,31,34)(10,25,12,26)(15,23)(16,29)(17,24)(18,35)(19,32,33,30)(21,28)(22,36);(2,4)(5,9,11,31)(6,15,18,20)(7,14,23,35)(8,13,27,34)(10,32,12,30)(16,28,17,36)(19,25,33,26)(21,24,22,29)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=ef,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 341264);