#501094

order	512
nilpotency class	4
isoclinism	isoclinic class of #501090
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×D₄	C₂⁴
prime radical	#23908	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1160 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1160 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)(7,12)(8,19)(9,14,23,11)(10,15)(13,21,31,25)(16,33)(17,28,24,26)(18,29)(20,36)(22,32)(27,34,30,35);(1,2)(3,4)(6,11)(9,23)(12,26)(13,19)(14,15)(17,24)(21,25)(27,33)(28,29)(30,36)(31,32)(34,35);(2,4)(5,7,10,18)(6,12,15,29)(8,20,22,16)(9,24,23,17)(11,26,14,28)(13,30,31,27)(19,36,32,33)(21,34,25,35);(5,8,10,22)(6,13,15,31)(7,16,18,20)(9,25,23,21)(11,19,14,32)(12,27,29,30)(17,35,24,34)(26,33,28,36)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^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,d^c=di,e^c=eh,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, 408290);