#376141

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1534 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283337);