#385894

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1455 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293090);