#346276

order	512
nilpotency class	3
isoclinism	isoclinic class of #346276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 965 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) = 32

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=def,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	#72235403, #72235523, #72284098, …
Fitting subgroup of	#72241385, #72252369, #72283899, …
maximal subgroup of	#72209707, #72228236, #72235378, …
maximal normal subgroup of	#72209707, #72228236, #72235378, …
maximal quotient of	#72209704, #72210067, #72210069, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253472);