#4435090

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 8388608
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2660 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)(10,13)(11,12)(17,18,23,26)(19,25,20,24)(21,31,32,22)(27,28,29,30);(9,10,14,13)(11,16,12,15)(17,19,23,20)(18,24,26,25)(21,27,32,29)(22,28,31,30);(5,6,8,7)(9,11,14,12)(10,15,13,16)(17,20,23,19)(18,25,26,24)(21,29,32,27)(22,30,31,28);(1,2,3,4)(5,7,8,6)(9,12,14,11)(10,15,13,16)(17,21,23,32)(18,22,26,31)(19,27,20,29)(24,28,25,30);(9,13,14,10)(11,16,12,15)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,29)(26,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4342286);