#285342

order	512
nilpotency class	3
isoclinism	isoclinic class of #285243
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

624 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192538);