#403353

order	512
nilpotency class	3
isoclinism	isoclinic class of #403352
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2433 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 2433 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

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, 310549);