#368897

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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

1656 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)(5,6,12,16,11,17,10,26)(7,18,20,13,21,31,27,30)(8,14,23,32,9,24,25,15)(19,29,36,28,22,33,35,34);(5,7,12,20,11,21,10,27)(6,13,16,31,17,30,26,18)(8,22,23,35,9,19,25,36)(14,28,32,33,24,34,15,29);(1,2,3,4)(5,8,11,9)(6,14,17,24)(7,19,21,22)(10,23,12,25)(13,28,30,34)(15,26,32,16)(18,33,31,29)(20,35,27,36);(5,9,11,8)(6,15,17,32)(7,19,21,22)(10,23,12,25)(13,29,30,33)(14,26,24,16)(18,34,31,28)(20,36,27,35)

Polycyclic presentation:

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

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, 276093);