#202434

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

812 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:

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

Polycyclic presentation:

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

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, 109630);