#469832

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16664
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,14)(4,20)(5,11)(6,22)(7,12)(8,13)(9,17)(10,23)(15,30)(16,21)(18,28)(19,27)(24,25)(26,32)(29,31);(1,3,6,17,8,19,7,18)(2,9,12,14,13,28,22,27)(4,21,23,32,24,31,11,15)(5,16,25,30,10,29,20,26);(1,4,8,24)(2,10,13,5)(3,15)(6,11,7,23)(9,26)(12,25,22,20)(14,29)(16,27)(17,31)(18,21)(19,32)(28,30);(1,5)(2,11,13,23)(3,16)(4,22,24,12)(6,25)(7,20)(8,10)(9,15,28,32)(14,21,27,31)(17,30)(18,26)(19,29)

Polycyclic presentation:

2^9|b^2=e,c^2=ef,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377028);