#5140948

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

2773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=ghi,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	#72300725, #72300749, #72300770
maximal normal subgroup of	#72300725, #72300749, #72300770

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5048144);