#3995210

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1693 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=gi,c^2=fgh,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3902406);