#228232

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#3480	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,d^c=dhi,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135428);