#651003

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

2880 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)(7,8)(9,10)(11,12)(13,14,18,17)(15,20,19,16)(21,22)(23,27)(24,28)(25,26);(6,10)(8,11)(22,26)(23,28);(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,20)(14,19,17,15)(21,24)(22,23)(25,27)(26,28);(5,8,9,11)(6,7,10,12)(13,17,18,14)(15,20,19,16)(21,22)(23,24)(25,26)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 558199);