#254298

order	512
nilpotency class	3
isoclinism	isoclinic class of #254275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

848 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161494);