#4035838

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2804 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)(6,10)(7,11)(14,17)(15,19)(21,22,24,26)(23,27,28,25);(5,6)(7,12)(8,11)(9,10)(15,19)(18,20)(21,22,24,26)(23,27,28,25);(6,10)(8,12)(13,14,16,17)(15,18,19,20)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12);(5,8)(6,11)(7,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=g,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, 3943034);