#501122

order	512
nilpotency class	4
isoclinism	isoclinic class of #501120
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	#23910	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1143 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1143 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=f,c^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,e^b=ei,f^b=fhi,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, 408318);