#4600685

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4507881);