#252258

order	512
nilpotency class	3
isoclinism	isoclinic class of #252247
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2404
prime radical	#18126	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

816 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:

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

Polycyclic presentation:

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

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, 159454);