#419721

order	512
nilpotency class	3
isoclinism	isoclinic class of #417661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326917);