#110392

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17588);