#518705

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5994 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)(13,14)(15,20)(16,24)(17,21)(18,23)(19,25)(22,26)(27,28);(9,10,11,12)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16,17,25)(14,19,21,24)(15,22,23,28)(18,26,20,27);(13,15)(14,20)(16,22)(17,23)(18,21)(19,27)(24,26)(25,28);(13,14)(15,18)(16,19)(17,21)(20,23)(22,26)(24,25)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 425901);