#183690

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

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,d^a=dhi,d^b=di,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	#72286284, #72286485, #72286509, …
maximal normal subgroup of	#72286284, #72286485, #72286509, …
maximal quotient of	#72321653, #72321765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90886);