#396375

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	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1455 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,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, 303571);