#215056

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:

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

Polycyclic presentation:

2^9|a^2=g,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, 122252);