#365994

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1129 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:

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

Polycyclic presentation:

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

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, 273190);