#698361

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2684 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 605557);