#466409

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

913 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fg,c^2=ei,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,e^d=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, 373605);