#183470

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dhi,c^b=ci,e^b=eh,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90666);