#498576

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
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×#2250	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

683 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)(5,6,10,16,33,17,28,34,12,19,27,14,11,18,8,25)(7,20,21,26,22,9,15,30,24,36,35,29,23,31,32,13);(1,3,4,2)(5,7)(6,13)(8,21)(9,14)(10,32)(11,22)(12,24)(15,27)(16,31)(17,29)(18,26)(19,30)(20,25)(23,33)(28,35)(34,36);(5,8,11,27,12,28,33,10)(6,14,17,25,19,16,18,34)(7,21,22,15,24,35,23,32)(9,29,20,30,31,26,36,13);(5,9,8,26,11,20,27,13,12,31,28,29,33,36,10,30)(6,15,25,22,18,21,14,7,19,32,34,23,17,35,16,24)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=eh,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,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405772);