#501123

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	#23909	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1127 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1127 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 408319);