#252220

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59228	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

964 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)(4,6)(5,8)(9,10)(11,12)(13,15)(14,16)(17,18,20,22)(19,23,26,29)(21,24,27,30)(25,32,28,31);(1,3,4,8)(2,5,6,7)(9,11,14,15)(10,12,16,13)(18,22)(19,25)(21,27)(23,31)(26,28)(29,32);(10,11)(12,13)(15,16)(17,19,20,26)(18,23,22,29)(21,28,27,25)(24,31,30,32);(2,6)(5,7)(9,12,14,13)(10,11,16,15)(18,22)(23,29)(24,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159416);