#153199

order	512
nilpotency class	5
isoclinism	isoclinic class of #153199
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13348
commutator, Frattini, cosocle	#532	C₂³
prime radical	#33570	C₂
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,f^a=fghi,g^a=gi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60395);