#417545

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
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
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2418 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)(7,15)(9,20)(11,24)(13,21)(16,32)(17,22)(18,23)(25,36)(26,29)(27,30)(31,34)(33,35);(1,2)(3,4)(5,6,10,14)(7,11,17,26)(8,12,19,28)(9,21,20,13)(15,29,22,24)(16,33,34,25)(18,30,23,27)(31,35,32,36);(5,7)(6,11)(8,16)(9,22)(10,17)(12,25)(13,29)(14,26)(15,20)(18,31)(19,34)(21,24)(23,32)(27,35)(28,33)(30,36);(2,4)(5,8)(6,12)(7,16)(9,23)(10,19)(11,25)(13,30)(14,28)(15,31)(17,34)(18,20)(21,27)(22,32)(24,35)(26,33)(29,36)

Polycyclic presentation:

2^9|b^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,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, 324741);