#358329

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#383
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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:

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

Polycyclic presentation:

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

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, 265525);