#5478656

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

3156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,c^2=g,e^2=gh,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	#72328131, #72328191, #72328346
maximal normal subgroup of	#72328131, #72328191, #72328346

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5385852);