#245117

order	512
nilpotency class	3
isoclinism	isoclinic class of #245105
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	#36957	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1238 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,11,16,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(1,3)(2,7)(4,10)(5,6)(8,15)(9,13)(11,16)(12,14)(18,23)(22,30)(24,31)(29,32)

Polycyclic presentation:

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

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, 152313);