#449969

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15954
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1008 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357165);