#338305

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24626	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

748 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,28,32,31);(1,3,5,2)(4,7,8,6)(17,19,21,27)(18,23,22,25)(20,28,29,31)(24,26,30,32);(1,4,5,8)(2,6,3,7)(10,13)(11,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32);(2,3)(6,7)(10,13)(11,16)(14,15)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,28,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=ei,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 245501);