#375947

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14172
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,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, 283143);