#202440

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109636);