#233561

order	512
nilpotency class	3
isoclinism	isoclinic class of #233559
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#37440	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1641 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285462
maximal normal subgroup of	#72285462

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140757);