#457681

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16299
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

866 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)(3,7)(4,15,17,28)(5,10)(6,9)(8,18,25,20)(11,13)(12,29,16,32)(14,23)(19,27)(21,26)(22,30,24,31);(1,2)(3,11,13,7)(4,8)(5,10)(6,9)(12,24,16,22)(14,27,19,23)(15,18)(17,25)(20,28)(21,26)(29,30,32,31);(1,3)(2,7)(4,16,17,12)(5,19)(6,13)(8,24,25,22)(9,11)(10,27)(14,21)(15,29,28,32)(18,30,20,31)(23,26);(1,4)(2,8)(3,12)(5,20)(6,17)(7,22)(9,25)(10,28)(11,24)(13,16)(14,31)(15,26)(18,21)(19,30)(23,32)(27,29)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi,f^d=fi,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, 364877);