#153195

order	512
nilpotency class	5
isoclinism	isoclinic class of #153131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2199
commutator, Frattini, cosocle	#532	C₂³
socle	C₂	#13348
prime radical	#33550	C₂
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 501 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60391);