#499853

order	512
nilpotency class	4
isoclinism	isoclinic class of #499849
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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	C₂×#3018	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1929 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1929 conjugacy classes of subgroups — too many to draw. Only its 150 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)(9,11)(10,15)(13,22)(14,23)(16,32)(17,26)(18,28)(20,29)(21,36)(24,31)(25,34)(27,33)(30,35);(1,2)(3,4)(6,11)(8,20)(9,23)(12,26)(13,29)(14,15)(16,21)(17,34)(19,31)(25,28)(27,36)(30,32);(2,4)(5,7,10,18)(6,12,15,28)(8,21,20,16)(9,17,23,34)(11,25,14,26)(13,30,31,27)(19,32,29,36)(22,35,24,33);(5,8,10,20)(6,13,15,31)(7,16,18,21)(9,24,23,22)(11,19,14,29)(12,27,28,30)(17,35,34,33)(25,36,26,32)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=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, 407049);