#215102

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
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	#41954	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,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=ce,d^b=di,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, 122298);