#357707

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2658 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)(5,6,10,16,11,15,9,14)(7,12,19,29,20,28,18,27)(8,21,23,32,25,13,24,31)(17,33,22,36,34,26,35,30);(1,2,3,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,30)(14,27)(15,28)(16,29)(17,24)(21,36)(23,34)(25,35)(26,32)(31,33);(5,8)(6,13)(7,17)(9,23)(10,24)(11,25)(12,26)(14,31)(15,21)(16,32)(18,22)(19,35)(20,34)(27,30)(28,33)(29,36);(5,9,11,10)(6,14,15,16)(7,18,20,19)(8,23,25,24)(12,27,28,29)(13,31,21,32)(17,22,34,35)(26,30,33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,g^c=gi,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, 264903);