#362869

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=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, 270065);