#419196

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1855 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)(5,6,9,24,10,15,27,14)(7,16,18,11,19,13,26,29)(8,20,22,31,23,12,25,32)(17,30,34,28,21,33,35,36);(5,7,9,18,10,19,27,26)(6,11,14,16,15,29,24,13)(8,21,22,35,23,17,25,34)(12,28,31,30,20,36,32,33);(2,4)(5,8,10,23)(6,12,15,20)(7,17,19,21)(9,25,27,22)(11,28,29,36)(13,30,16,33)(14,32,24,31)(18,35,26,34);(5,7)(6,13)(8,17)(9,26)(10,19)(11,24)(12,30)(14,29)(15,16)(18,27)(20,33)(21,23)(22,35)(25,34)(28,31)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326392);