#232181

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

539 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,e^a=eh,c^b=cefi,d^b=de,f^b=fh,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139377);