#93249

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₈×C₁₆ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₄²
commutator	C₄	C₈×C₁₆
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 264 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 40 points:

(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22)(25,26,27,30,28,31,33,36,29,32,34,37,35,38,39,40);(1,2,3,5,4,6,7,8)(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,b^a=bc,e^a=ef,d^b=df

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210241, C₂×#93249, #72210645
maximal normal subgroup of	#72210241, C₂×#93249, #72210645
maximal quotient of	C₂×#93249, #72210629, #72210635, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 445);