#183611

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

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	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90807);