#102037

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#493	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

546 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=eh,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210554, #72210562
maximal normal subgroup of	#72210554, #72210562
maximal quotient of	#72210318, #72210332

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9233);