#94633

order	512
nilpotency class	6
isoclinism	isoclinic class of #94630
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7394
commutator	#407	C₂×C₄
Frattini, cosocle	#1784	C₂²
prime radical	C₂⁴	#102
LCS	G ▷ #407 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1784 ◁ G
LECS	G ▷ #1784 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 22 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fi,d^2=i,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=defi,e^b=ei,f^b=fi,g^b=gh,d^c=dgi,e^c=ehi,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1829);