#304975

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

743 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=efi,d^2=eh,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=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212171);