#204606

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1280 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111802);