#401070

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1692 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)(7,12,19,28)(8,13)(9,22,17,30)(10,16)(11,15)(14,23,33,36)(18,32)(20,29,24,31)(21,27)(25,35)(26,34);(1,2)(3,4)(5,7)(6,12,15,28)(8,17)(9,18)(10,24)(11,19)(13,30,32,22)(14,27,33,35)(16,31,34,29)(20,26)(21,23)(25,36);(2,4)(5,8,11,18)(6,13,15,32)(7,17,19,9)(10,25,26,21)(12,22,28,30)(14,31,33,29)(16,35,34,27)(20,23,24,36);(5,9,11,17)(6,14)(7,18,19,8)(10,23,26,36)(12,27)(13,31)(15,33)(16,22)(20,25,24,21)(28,35)(29,32)(30,34)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=efi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,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, 308266);