#227147

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
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	#18510	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134343);