#139481

order	512
nilpotency class	4
isoclinism	isoclinic class of #139432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251902, #72252003, #72252005
maximal normal subgroup of	#72251902, #72252003, #72252005

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46677);