#283834

order	512
nilpotency class	3
isoclinism	isoclinic class of #283771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 191030);