#615567

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
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

2628 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522763);