#501143

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#3487	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408339);