#184230

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 91426);