#5024595

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 2097152
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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2206 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=fgi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4931791);