#190020

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1037 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 1037 conjugacy classes of subgroups — too many to draw. Only its 276 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,g^b=gh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97216);