#121113

order	512
nilpotency class	3
isoclinism	isoclinic class of #121113
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

511 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi

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, 28309);