#437013

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15579
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 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)(3,8)(4,15,22,23)(5,10)(7,27,13,11)(9,28,31,21)(14,24)(16,30)(17,29)(18,20)(19,26)(25,32);(1,3,13,18,6,16,7,17)(2,8,11,20,12,30,27,29)(4,19,21,14,22,32,28,10)(5,23,26,31,24,15,25,9);(1,4)(2,9)(3,14)(5,20)(6,22)(7,28)(8,25)(10,16)(11,23)(12,31)(13,21)(15,27)(17,19)(18,32)(24,29)(26,30);(1,5,13,26,6,24,7,25)(2,10,11,19,12,14,27,32)(3,15,18,9,16,23,17,31)(4,20,21,30,22,29,28,8)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338916, #72338919, #72338927
maximal normal subgroup of	#72338916, #72338919, #72338927

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344209);