#362090

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
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

1270 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,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, 269286);