#4335641

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fhi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4242837);