#364443

order	512
nilpotency class	3
isoclinism	isoclinic class of #196344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2404 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)(5,6,9,14,10,13,23,29)(7,11,17,26,16,25,32,35)(8,18,20,30,21,12,22,28)(15,27,31,36,19,24,33,34);(2,4)(5,7,9,17,10,16,23,32)(6,11,14,26,13,25,29,35)(8,19,20,33,21,15,22,31)(12,27,28,36,18,24,30,34);(5,8,10,21)(6,12,13,18)(7,15,16,19)(9,22,23,20)(11,24,25,27)(14,30,29,28)(17,33,32,31)(26,36,35,34);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271639);