#495537

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21395
prime radical	#60693	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

836 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=fh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214311, #72214675, #72214702, …
maximal normal subgroup of	#72214311, #72214675, #72214702, …
maximal quotient of	#72226840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402733);