#251460

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

933 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)(10,14)(11,13)(17,18,22,26)(19,27,29,20)(21,32,23,25)(24,31,28,30);(1,3,4,8)(2,5,6,7)(9,10,12,14)(11,16,13,15)(17,19,22,29)(18,20,26,27)(21,28,23,24)(25,30,32,31);(10,15)(11,13)(14,16)(17,20,22,27)(18,24,26,28)(19,25,29,32)(21,31,23,30);(2,6)(5,7)(9,11,12,13)(10,16,14,15)(17,21,22,23)(18,25,26,32)(19,28,29,24)(20,30,27,31)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

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, 158656);