#4389779

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2760 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)(6,7)(10,14)(12,15)(17,18,23,28)(19,25,20,24)(21,22,27,26)(29,30,31,32);(9,10,13,14)(11,15,16,12)(17,19,23,20)(18,24,28,25)(21,31,27,29)(22,30,26,32);(5,6,8,7)(10,14)(12,15)(17,20,23,19)(18,25,28,24)(21,31,27,29)(22,30,26,32);(1,2,3,4)(5,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,21,23,27)(18,26,28,22)(19,29,20,31)(24,30,25,32);(9,12)(10,11)(13,15)(14,16)(17,22,23,26)(18,27,28,21)(19,30,20,32)(24,29,25,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=g,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4296975);