#5160584

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=h,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232617, #72232622, #72232640
maximal normal subgroup of	#72232617, #72232622, #72232640

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5067780);