#203587

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dh,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, 110783);