#193424

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 100620);