#245272

order	512
nilpotency class	3
isoclinism	isoclinic class of #245246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#29634	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1463 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,23)(20,24)(21,29)(22,25)(26,32)(27,30)(28,31);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,22,21)(18,23,25,29)(20,26,28,27)(24,30,31,32);(6,12)(13,16)(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,31)(29,32);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,22,19)(18,23,25,29)(20,27,28,26)(24,30,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284232, #72284327, #72284452, …
maximal normal subgroup of	#72284232, #72284327, #72284452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152468);