#402208

order	512
nilpotency class	3
isoclinism	isoclinic class of #397903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2856
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14836
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309404);