#214927

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#42881	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,f^c=fi,g^d=gh

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, 122123);