#246331

order	512
nilpotency class	3
isoclinism	isoclinic class of #246319
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	#46356	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1247 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:

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

Polycyclic presentation:

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

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, 153527);