#899380

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2882 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)(7,11)(10,12)(13,14,17,16)(15,19,20,18)(21,22,26,25)(23,28,27,24);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(14,16)(18,19)(23,27)(24,28);(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,19,20,18)(21,24,26,28)(22,27,25,23);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,15)(14,18)(16,19)(17,20)(21,25,26,22)(23,28,27,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 806576);