#193661

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	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

1204 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)(6,10,12,15)(7,13,14,16)(21,22,25,29,26,30,24,28)(23,31,34,27,32,36,33,35);(1,2,3,4)(5,6,8,15,9,12,17,10)(7,11,16,20,14,19,13,18)(21,23)(22,27)(24,34)(25,33)(26,32)(28,36)(29,31)(30,35);(5,7)(6,11)(8,16)(9,14)(10,18)(12,19)(13,17)(15,20)(21,23)(22,27)(24,34)(25,33)(26,32)(28,36)(29,31)(30,35);(6,12)(10,15)(11,19)(18,20)(23,32)(27,35)(31,36)(33,34)

Polycyclic presentation:

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

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, 100857);