#215085

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	#35844	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

954 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)(3,7,4,9)(5,11)(6,8)(10,14)(12,15,13,16)(17,18,21,24,22,25,20,29)(19,26,28,23,27,32,30,31);(1,3,5,4)(2,7,11,9)(6,13,14,12)(8,16,10,15)(19,27)(23,31)(26,32)(28,30);(2,8)(6,14)(7,15)(9,16)(10,11)(12,13)(17,19,22,27)(18,23,25,31)(20,30,21,28)(24,32,29,26);(1,4,5,3)(2,9,11,7)(6,13,14,12)(8,16,10,15)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=h,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, 122281);