#358331

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,e^2=i,f^2=i,h^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, 265527);