#229581

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	#2361
prime radical	#3106	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi

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, 136777);