#213922

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36702	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1366 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1366 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121118);