#371187

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14068
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1127 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278383);