#220682

order	512
nilpotency class	3
isoclinism	isoclinic class of #220548
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61215	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

899 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127878);