#112275

order	512
nilpotency class	3
isoclinism	isoclinic class of #112268
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1053 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1053 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19471);