#392032

order	512
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2840
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14673
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1520 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1520 conjugacy classes of subgroups — too many to draw. Only its 232 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,h^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262960
maximal normal subgroup of	#72262960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299228);