#416455

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323651);