#248246

order	512
nilpotency class	3
isoclinism	isoclinic class of #248167
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1070 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)(17,18,22,26)(19,27,28,31)(20,23,21,25)(24,30,32,29);(2,6)(3,11)(4,13)(7,15)(8,9)(10,16)(18,23)(19,28)(20,21)(24,31)(25,26)(27,32);(17,19,22,28)(18,24,26,32)(20,29,21,30)(23,31,25,27);(1,3,5,11)(2,7,9,16)(4,14,13,12)(6,10,8,15)(17,19)(18,24)(20,30)(21,29)(22,28)(23,27)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155442);