#6436633

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62686	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=dh,c^b=ch,g^b=gi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317879
maximal normal subgroup of	#72317879

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343829);