#368400

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1328 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)(5,6,10,15,11,16,9,26)(7,17,21,28,20,13,19,12)(8,22,24,30,25,36,23,14)(18,32,35,29,34,27,33,31);(1,2,4,3)(5,7)(6,12,16,28)(8,18)(9,19)(10,21)(11,20)(13,15,17,26)(14,27,30,32)(22,31,36,29)(23,33)(24,35)(25,34);(5,7)(6,13,16,17)(8,18)(9,19)(10,21)(11,20)(12,26,28,15)(14,29,30,31)(22,27,36,32)(23,33)(24,35)(25,34);(1,3,4,2)(5,8,9,23,11,25,10,24)(6,14,15,22,16,30,26,36)(7,18,19,33,20,34,21,35)(12,27,17,31,28,32,13,29)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 275596);