#285420

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	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

635 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,d^2=h,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, 192616);