#353695

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260891);