#228202

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
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	#3475	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

669 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,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, 135398);