#189969

order	512
nilpotency class	3
isoclinism	isoclinic class of #161128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1282 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1282 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257647, #72257857
maximal normal subgroup of	#72257647, #72257857

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97165);