#471023

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16735
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

630 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378219);