#183421

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

770 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)(5,6,9,14,10,15,8,13)(7,12,16,11,18,19,17,20)(21,22,24,28,26,30,25,29)(23,31,33,27,34,35,32,36);(5,7)(6,11)(8,16)(9,17)(10,18)(12,14)(13,19)(15,20)(21,23,25,33,26,34,24,32)(22,27,29,35,30,36,28,31);(1,3,4,2)(5,7)(6,12)(8,17)(9,16)(10,18)(11,14)(13,20)(15,19);(5,8,10,9)(6,13,15,14)(7,16,18,17)(11,19,20,12)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,31,36,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,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, 90617);