#355493

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1463 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:

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

Polycyclic presentation:

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

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, 262689);