#204372

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111568);