#417572

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2417 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:

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

Polycyclic presentation:

2^9|d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 324768);