#121014

order	512
nilpotency class	3
isoclinism	isoclinic class of #121012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

426 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,10)(4,12,14,11)(5,8)(6,9)(7,13,15,16)(17,18,23,28,21,26,22,27)(19,20,25,30,29,24,31,32);(1,3,5,2,6,8,10,9)(4,12,13,15,14,11,16,7)(17,19)(18,24,26,20)(21,29)(22,25)(23,31)(27,32,28,30);(1,4,5,13,6,14,10,16)(2,7,3,11,9,15,8,12)(17,20,21,24)(18,25)(19,28)(22,30,23,32)(26,31)(27,29)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,g^b=gi,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28210);