#94382

order	512
nilpotency class	5
isoclinism	isoclinic class of #94303
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1393
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	#2929	C₂²
socle	C₂	#7283
prime radical	#525	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #2929 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

230 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 42 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=fh,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, 1578);