#203607

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

700 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)(3,4)(5,6,9,12)(7,11,10,8)(13,14,18,21,17,20,26,27)(15,22,24,28,23,16,19,25);(2,4)(5,7,9,10)(6,8,12,11)(13,15,18,24,17,23,26,19)(14,16,21,25,20,22,27,28);(5,8,10,6,9,11,7,12)(13,16)(14,19)(15,21)(17,22)(18,25)(20,24)(23,27)(26,28);(5,7,9,10)(6,11,12,8)(15,23)(16,22)(19,24)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=di,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110803);