#148823

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	#1889
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56019);