#147339

order	512
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×D₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1889
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

703 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 154 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54535);