#346293

order	512
nilpotency class	3
isoclinism	isoclinic class of #346284
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

772 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,9,11,6)(4,12,13,7)(10,15,16,14)(22,30)(31,32);(1,3)(2,6,8,9)(4,10)(5,11)(7,15,12,14)(13,16)(17,18)(19,22)(20,27,29,23)(21,24)(25,31,28,32)(26,30);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(17,19)(18,22)(20,28,29,25)(21,26)(23,32,27,31)(24,30);(3,11)(6,9)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,31,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=defh,e^b=eh,f^b=fhi,d^c=de,e^c=ei,f^c=fh,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72340908, #72341183
Fitting subgroup of	#72340908, #72340910, #72341073
maximal subgroup of	#72283693, #72283755, #72284366, …
maximal normal subgroup of	#72283693, #72283755, #72284366, …
maximal quotient of	#72240079

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253489);