#5478654

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

2617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300343, #72300400, #72300421
maximal normal subgroup of	#72300343, #72300400, #72300421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5385850);