#387440

order	512
nilpotency class	3
isoclinism	isoclinic class of #31282
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14473
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1157 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338914, #72338916, #72338924
maximal normal subgroup of	#72338914, #72338916, #72338924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294636);