#94069

order	512
nilpotency class	5
isoclinism	isoclinic class of #7338
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1316
commutator	C₂⁴	C₄×C₈
Frattini, cosocle	#2280	C₂²
socle	C₂	#7187
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2280 ◁ G
LECS	G ▷ #2280 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,e^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,d^b=df,g^b=gi,d^c=dh,f^d=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, 1265);