#233617

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279567, #72279890, #72280045, …
maximal normal subgroup of	#72279567, #72279890, #72280045, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140813);