#469729

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16661
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

949 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)(4,15,17,31)(5,10)(6,9)(8,18,16,30)(11,13)(12,26,27,29)(14,22)(19,24)(20,23)(21,28,25,32);(1,3)(2,7)(4,12)(5,19)(6,13)(8,21)(9,11)(10,24)(14,20)(15,29)(16,25)(17,27)(18,32)(22,23)(26,31)(28,30);(1,2)(3,11,13,7)(4,16)(5,10)(6,9)(8,17)(12,21,27,25)(14,24,19,22)(15,30)(18,31)(20,23)(26,32,29,28);(1,4)(2,8,9,16)(3,12)(5,18)(6,17)(7,21,11,25)(10,15,23,31)(13,27)(14,28)(19,32)(20,30)(22,26,24,29)

Polycyclic presentation:

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

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, 376925);