#419821

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1645 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)(7,16,18,11)(8,12)(9,15)(10,14)(13,19,31,22)(17,25,33,26)(20,29)(21,28)(23,32)(24,30)(27,34,36,35);(1,2)(3,4)(5,7)(6,11)(8,17)(9,22)(10,18)(12,26)(13,30)(14,16)(15,31)(19,24)(20,35)(21,33)(23,34)(25,28)(27,32)(29,36);(2,4)(5,8,10,21)(6,12,14,28)(7,17,18,33)(9,23,24,20)(11,25,16,26)(13,27,31,36)(15,32,30,29)(19,35,22,34);(5,7)(6,13,14,31)(8,17)(9,19)(10,18)(11,15,16,30)(12,27,28,36)(20,34)(21,33)(22,24)(23,35)(25,32,26,29)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,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, 327017);