#181602

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1728 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)(6,10)(7,8)(11,12)(13,14)(15,22)(16,20)(17,25)(18,21)(19,24)(23,28)(26,27);(5,6,8,12,9,11,7,10)(13,15)(14,19)(16,24)(17,26)(18,23)(20,28)(21,27)(22,25);(6,11)(10,12)(15,23)(19,27)(22,28)(24,26);(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,g^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

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, 88798);