#118369

order	512
nilpotency class	3
isoclinism	isoclinic class of #118368
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25565);