#441642

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 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:

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

Polycyclic presentation:

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

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, 348838);