#150825

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

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

normal subgroups only

This group has 365 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)(3,14,18,8)(4,20,19,28)(6,13,24,27)(7,12,17,26)(9,22,10,29)(15,32,21,25)(16,31,23,30);(2,8)(3,15)(4,21)(5,23)(6,16)(9,26)(10,13)(11,28)(12,25)(14,27)(17,24)(18,22)(20,31)(30,32);(1,3,16,29,7,15,6,19)(2,9,13,14,12,28,31,32)(4,17,18,24,22,5,21,23)(8,11,20,27,25,26,10,30)

Polycyclic presentation:

2^9|a^2=e,c^2=ghi,d^2=gi,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, 58021);