#293866

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28448	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=dhi,g^c=gi

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, 201062);