#204379

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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	C₂×#449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1063 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)(7,16)(8,12)(9,14)(10,19)(11,15)(13,18)(17,20)(21,22,25,29,26,30,24,27)(23,31,33,36,32,35,34,28);(1,2,3,4)(5,7,10,8,11,18,9,17)(6,12,14,16,15,20,19,13)(22,27,30,29)(23,32)(28,31,36,35)(33,34);(5,8,10,18,11,17,9,7)(6,13,14,12,15,16,19,20)(21,23,24,34,26,32,25,33)(22,28,29,31,30,36,27,35);(5,9,11,10)(6,14,15,19)(7,17,18,8)(12,16,20,13)(21,24,26,25)(22,27,30,29)(23,33,32,34)(28,31,36,35)

Polycyclic presentation:

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

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, 111575);