#243758

order	512
nilpotency class	3
isoclinism	isoclinic class of #233100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 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)(3,6)(4,5)(7,8)(9,10,14,15,13,11,12,16)(17,18,20,24,21,25,29,23)(19,22,26,30,28,32,27,31);(2,5)(6,8)(10,11)(15,16)(17,19,21,28)(18,22,25,32)(20,26,29,27)(23,31,24,30);(9,11,13,10)(12,16,14,15)(18,23)(19,26)(20,29)(24,25)(27,28)(30,31);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,11,16)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150954);