#204130

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
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	C₂²×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1222 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:

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

Polycyclic presentation:

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

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, 111326);