#150827

order	512
nilpotency class	5
isoclinism	isoclinic class of #150824
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11877
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#28497	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

363 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efg,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58023);