#405581

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,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, 312777);