#499947

order	512
nilpotency class	4
isoclinism	isoclinic class of #499946
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

956 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

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

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, 407143);