#319542

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#33175	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224779, #72224781, #72224786
maximal normal subgroup of	#72224779, #72224781, #72224786

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226738);