#193474

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,15,17,20)(8,12)(9,14)(10,13)(11,16,19,18)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(5,7,9,18,10,17,8,16)(6,11,14,20,13,19,12,15)(21,23,24,30,25,29,31,35)(22,26,27,33,28,32,34,36);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,15,19,20);(6,13)(11,19)(12,14)(15,20)(23,29)(26,32)(30,35)(33,36)

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,d^b=di,e^b=eh,g^b=gh

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, 100670);