#94081

order	512
nilpotency class	5
isoclinism	isoclinic class of #7349
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1317
commutator	C₂⁴	C₄×C₈
Frattini, cosocle	C₂×#407	C₂²
socle	C₂	#7187
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂×#407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 402 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,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, 1277);