#506242

order	512
nilpotency class	4
isoclinism	isoclinic class of #499949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

697 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,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,d^b=di,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, 413438);