#121116

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

526 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 108 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)(3,6,10,14)(4,11)(5,8)(7,12)(9,13,16,15)(17,18,22,26,23,25,21,20)(19,27,30,24,29,31,28,32);(1,3)(2,6,8,14)(4,9,12,16)(5,10)(7,15)(11,13)(17,19,21,30,23,29,22,28)(18,24,20,31,25,32,26,27);(1,4)(2,7,8,11)(3,9,10,16)(5,12)(6,13)(14,15)(17,20,21,25,23,26,22,18)(19,27,28,32,29,31,30,24)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,g^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, 28312);