#233553

order	512
nilpotency class	3
isoclinism	isoclinic class of #233549
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#37436	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251101, #72251150, #72251151, …
maximal normal subgroup of	#72251101, #72251150, #72251151, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140749);