#139452

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

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

normal subgroups only

This group has 622 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238635, #72238638, #72238641
maximal normal subgroup of	#72238635, #72238638, #72238641

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46648);