#517117

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2560 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:

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

Polycyclic presentation:

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

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, 424313);