#539856

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2278 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=fi,e^2=fh,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, 447052);