#6393701

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#537	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,e^c=ei

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, 6300897);