#215074

order	512
nilpotency class	3
isoclinism	isoclinic class of #214736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122270);