#416412

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

908 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,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, 323608);