#183389

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

846 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:

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

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,c^a=ch,d^a=dhi,e^b=eh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90585);