#355489

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
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	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1467 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=i,f^2=i,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^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, 262685);