#354933

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262129);