#122828

order	512
nilpotency class	4
isoclinism	isoclinic class of #122704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂³	C₂×#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

283 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=g,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cfh,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30024);