#394542

order	512
nilpotency class	3
isoclinism	isoclinic class of #394022
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,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,g^b=gi,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, 301738);