#263327

order	512
nilpotency class	3
isoclinism	isoclinic class of #263311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 839 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306295, #72306311, #72306320
maximal normal subgroup of	#72306295, #72306311, #72306320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170523);