#615492

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2443 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)(6,8)(10,11)(14,18)(15,16)(21,22)(23,25)(24,27)(26,28);(1,2,4,3)(5,6,9,8)(7,10,12,11)(13,14)(15,19)(16,20)(17,18)(22,23)(26,27);(1,2,4,3)(13,15,17,16)(14,19,18,20)(21,23,25,22)(24,27,28,26);(5,7,9,12)(6,10,8,11)(13,14)(15,19)(16,20)(17,18);(1,3,4,2)(5,8,9,6)(7,11,12,10)(13,16,17,15)(14,20,18,19)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^9|b^2=fhi,c^2=g,d^2=fg,e^2=fh,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, 522688);