#396120

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1430 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,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, 303316);