#4386404

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3670 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)(5,6,9,10)(7,11,12,8)(13,14,17,19)(15,16,20,18)(23,27)(26,28);(5,7,9,12)(6,8,10,11)(13,15,17,20)(14,18,19,16)(22,25)(26,28);(1,2)(3,4)(6,10)(7,12)(13,16,17,18)(14,15,19,20)(21,22)(23,26)(24,25)(27,28);(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19)(18,20);(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4293600);