#517893

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(5,6,7,8)(9,10,13,12)(11,14,15,16)(17,18,22,23)(19,24,26,20)(21,25,30,31)(27,29,32,28);(5,6,7,8)(11,15)(14,16)(18,23)(19,26)(25,31)(27,32);(5,6)(7,8)(9,11,13,15)(10,14,12,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,29)(28,31)(30,32);(1,2,3,4)(9,12,13,10)(11,16,15,14)(17,20,22,24)(18,19,23,26)(21,29,30,28)(25,32,31,27);(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fgh,d^2=i,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, 425089);