#498575

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,d^2=ehi,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405771);