#535580

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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

3730 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=h,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, 442776);