#372769

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1440 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279965);