#413086

order	512
nilpotency class	3
isoclinism	isoclinic class of #397240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1550 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320282);