#420458

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	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1691 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ei,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, 327654);