#4421475

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
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

2464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,3,4)(6,10)(8,11)(14,16)(15,20)(21,22,26,24)(23,29,31,27)(25,28,34,32)(30,36,33,35);(5,6,9,10)(7,8,12,11)(13,14,17,16)(15,19,20,18)(21,23)(22,27)(24,29)(25,30)(26,31)(28,35)(32,36)(33,34);(5,7)(6,11)(8,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,22)(23,29,31,27)(25,32,34,28)(30,36,33,35);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16,17,14)(15,19,20,18)(21,23)(22,27)(24,29)(25,33)(26,31)(28,36)(30,34)(32,35);(5,8)(6,12)(7,10)(9,11)(13,15)(14,19)(16,18)(17,20)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,36)(31,33)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=fh,d^2=g,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, 4328671);