#115089

order	512
nilpotency class	3
isoclinism	isoclinic class of #115084
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

793 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 793 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ehi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22285);