#535406

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 442602);