#402207

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1293 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)(5,6,10,14,11,15,9,24)(7,16,20,27,19,32,18,12)(8,13,22,30,23,31,25,29)(17,33,21,26,34,28,35,36);(5,7,9,18,11,19,10,20)(6,12,14,16,15,27,24,32)(8,21,22,34,23,35,25,17)(13,28,29,26,31,33,30,36);(5,8)(6,13)(7,17,19,34)(9,25)(10,22)(11,23)(12,26,27,36)(14,30)(15,31)(16,28,32,33)(18,35,20,21)(24,29);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309403);