#615585

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2890 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=i,d^2=g,e^2=f,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, 522781);