#230391

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

522 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,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, 137587);