#285402

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
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	#18568	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

642 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)(3,12,13,15)(4,14,5,8)(6,11,16,9)(17,18,20,25,22,26,21,24)(19,27,29,32,30,31,28,23);(1,3)(2,4)(5,10)(6,14)(7,13)(8,16)(9,15)(11,12)(17,19,21,28,22,30,20,29)(18,23,24,31,26,32,25,27);(1,4,7,5)(2,8,10,14)(3,9,13,11)(6,12,16,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,32)(30,31);(1,5,7,4)(2,3,10,13)(6,12,16,15)(8,9,14,11)(17,18)(19,23)(20,25)(21,24)(22,26)(27,29)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=hi,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=ghi,f^c=fi,g^c=gh,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, 192598);