#231740

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	#2366
prime radical	C₂³	#559
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

523 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:

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

Polycyclic presentation:

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

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, 138936);