#127744

order	512
nilpotency class	4
isoclinism	isoclinic class of #11992
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1555
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8040
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

391 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=i,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265706, #72265714
maximal normal subgroup of	#72265706, #72265714

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34940);