#203594

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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

658 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)(8,15)(10,16)(12,18)(14,19)(17,20)(21,22,26,23,25,28,33,30)(24,27,32,29,31,34,36,35);(2,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,23,26,28,25,30,33,22)(24,29,32,34,31,35,36,27);(5,7,8,13,9,14,15,19)(6,10,12,16,11,17,18,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36);(6,11)(7,14)(12,18)(13,19)(24,31)(27,34)(29,35)(32,36)

Polycyclic presentation:

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

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, 110790);