#4044694

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1908 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)(6,10)(7,8)(13,14,17,19)(15,16,20,18);(5,6,9,10)(7,12,8,11)(15,20)(16,18)(21,22,25,24)(23,27,28,26);(5,7,9,8)(6,11,10,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(21,24,25,22)(23,27,28,26);(5,6)(7,12)(8,11)(9,10)(13,16,17,18)(14,15,19,20)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fgh,d^2=fgh,e^2=fg,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, 3951890);