#401005

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1678 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,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, 308201);