#462157

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

691 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369353);