#238261

order	512
nilpotency class	3
isoclinism	isoclinic class of #238259
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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1079 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236944, #72236996, #72237004
maximal normal subgroup of	#72236944, #72236996, #72237004

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145457);