#392434

order	512
nilpotency class	3
isoclinism	isoclinic class of #190022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^b=hi,h^c=hi,g^d=gi

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, 299630);