#248610

order	512
nilpotency class	3
isoclinism	isoclinic class of #248599
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	C₂³×C₄	C₂⁴
prime radical	#24140	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,d^c=di,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, 155806);