#238306

order	512
nilpotency class	3
isoclinism	isoclinic class of #238303
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1019 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)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,32)(29,31);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32);(4,13)(7,11)(9,12)(14,16)(17,20,22,21)(18,25,26,24)(19,27,28,29)(23,30,32,31)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=hi,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,g^b=gh,d^c=di,e^c=eh,f^c=fhi

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, 145502);