#285429

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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=i,d^2=i,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, 192625);