#94304

order	512
nilpotency class	5
isoclinism	isoclinic class of #94303
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1393
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	C₂×#407	C₂²
socle	C₂	#7250
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂×#407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

392 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,g^b=gh,e^c=ehi,g^c=gi

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, 1500);