#416457

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323653);