#283682

order	512
nilpotency class	3
isoclinism	isoclinic class of #283643
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	#18870	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

742 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,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,e^b=eh,g^b=gi,d^c=di,f^c=fh,g^c=ghi,e^d=eh

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, 190878);