#449075

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15929
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

804 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)(3,7,12,23)(4,14)(5,18)(6,9)(8,16)(10,20)(11,27,15,32)(13,30,29,22)(17,26)(19,25)(21,31,24,28);(1,3,5,13,6,12,20,29)(2,7,10,22,9,23,18,30)(4,15,16,31,17,11,19,28)(8,24,14,32,25,21,26,27);(1,4)(2,8)(3,11)(5,19)(6,17)(7,21)(9,25)(10,26)(12,15)(13,31)(14,18)(16,20)(22,32)(23,24)(27,30)(28,29);(1,2)(3,7,12,23)(4,8)(5,18)(6,9)(10,20)(11,21,15,24)(13,30,29,22)(14,19)(16,26)(17,25)(27,28,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338480, #72338481, #72338492
maximal normal subgroup of	#72338480, #72338481, #72338492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356271);