#420075

order	512
nilpotency class	3
isoclinism	isoclinic class of #420074
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15274
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 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)(3,15,18,26)(4,19,23,30)(5,9,17,11)(6,12)(7,13)(8,14)(10,22,29,24)(16,25)(20,27)(21,32)(28,31);(1,3)(2,9)(4,20)(5,6)(7,17)(8,18)(10,27)(11,14)(12,26)(13,15)(16,22)(19,25)(21,23)(24,31)(28,30)(29,32);(1,4)(2,10)(3,16)(5,21)(6,22)(7,24)(8,23)(9,25)(11,28)(12,19)(13,30)(14,29)(15,27)(17,20)(18,31)(26,32);(1,5,7,18,8,17,6,3)(2,11,13,15,14,9,12,26)(4,21,24,31,23,20,22,16)(10,28,30,27,29,25,19,32)

Polycyclic presentation:

2^9|a^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327271);