#420459

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1693 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)(7,17)(8,10)(9,24,28,23)(11,15)(12,16)(13,20)(14,27,32,26)(18,22)(19,34,36,29)(21,30)(25,31,35,33);(1,2)(3,4)(5,7,12,22)(6,13,16,30)(8,17,15,18)(9,25,28,35)(10,21,11,20)(14,31,32,33)(19,27,36,26)(23,34,24,29);(2,4)(5,8,10,6,12,15,11,16)(7,18,20,13,22,17,21,30)(9,23,26,14,28,24,27,32)(19,33,25,29,36,31,35,34);(5,9,10,26,12,28,11,27)(6,14,15,24,16,32,8,23)(7,19,20,25,22,36,21,35)(13,29,17,31,30,34,18,33)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 327655);