#501333

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

683 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 683 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408529);