#4033855

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3710 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)(6,8)(14,15)(16,23)(17,25)(27,28);(2,4)(10,12)(13,14,18,15)(16,21,23,19)(17,22,25,20)(24,27,26,28);(1,2)(3,4)(9,10)(11,12)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,27,26,28);(5,6)(7,8)(13,16,18,23)(14,19,15,21)(17,26,25,24)(20,28,22,27);(1,2)(3,4)(13,17,18,25)(14,20,15,22)(16,24,23,26)(19,27,21,28)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,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, 3941051);