#4133776

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
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	#16250	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2883 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)(10,12)(11,15)(17,18,22,23)(19,20,25,28)(21,24,31,32)(26,27,29,30);(5,6,7,8)(10,12)(14,16)(18,23)(19,25)(24,32)(26,29);(5,6)(7,8)(9,10,13,12)(11,16,15,14)(17,19)(18,20)(21,29)(22,25)(23,28)(24,30)(26,31)(27,32);(1,2)(3,4)(9,11,13,15)(10,14,12,16)(17,20)(18,19)(21,30)(22,28)(23,25)(24,29)(26,32)(27,31);(9,12,13,10)(11,16,15,14)(17,21,22,31)(18,24,23,32)(19,26,25,29)(20,27,28,30)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=g,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4040972);