#376623

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

969 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ei,d^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 283819);