#149284

order	512
nilpotency class	4
isoclinism	isoclinic class of #13357
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂²×Q₈	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1867
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

696 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257195, #72257231, #72257236
maximal normal subgroup of	#72257195, #72257231, #72257236

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56480);