#396376

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1407 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,h^c=hi,f^d=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, 303572);