#376200

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1507 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=eh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283396);