#4029313

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=fgi,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3936509);