#183608

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

780 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)(5,6,9,11,10,13,8,14)(7,12,17,18,15,20,16,19)(21,22,25,29,26,30,24,28)(23,31,33,27,34,35,32,36);(6,11)(8,9)(12,18)(13,14)(16,17)(19,20)(21,23,24,33,26,34,25,32)(22,27,28,35,30,36,29,31);(1,3,4,2)(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19);(6,13)(7,15)(11,14)(16,17)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,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.

Relationships

maximal subgroup of	#72319070, #72319075, #72319106, …
maximal normal subgroup of	#72319070, #72319075, #72319106, …
maximal quotient of	#72322936

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90804);